My personal recommendations, mostly stuff I've got:
Soldering iron: I like the Hakko FX-888D. $90-110 or so. They have better if you can afford it, but that one's very good. The FX-951 is the next step up, and can take micro-soldering handpieces and has the quick-change tips. It's about $240.
Get a chisel tip, eg Hakko T18-S3, a bevel tip (T18-S6), and a bent-conical tip (T18-BR02). The conical tip is perfect for lots of general purpose work, you can use the fine point or the sides of the bend. The back of the bend can be used for drag soldering, the inside of the bend makes soldering wires together easy. The chisel tip is good for soldering things with more thermal mass (PCB-mount heatsinks) and the bevel tip is pretty necessary for drag soldering on QFP and similar surface mount packages.
Hot air station: Probably something cheap from china, there aren't any particularly affordable name-brand ones that I know of. Weller has the WHA900 for around $600.
Magnification: Get at least one of the magnifying headsets ($8-10 on Amazon) and a desk magnifier with LED ring light. Better option is an AmScope stereo microscope, such as the SM-4NTP and a ring light for it like the LED-144W-ZK. $480 total.
PCB vise: I have an Aven 17010, it works pretty well. MUCH better than trying to hold a board in the helping hands.
Flux: Get liquid flux with a syringe. Amtech is the best, but there is a lot of counterfeit stuff out there, and Amtech doesn't sell it directly (bulk orders only). https://mailin.repair/amtech-nc-559-v2-30-cc-16160.html sells the real flux.
Tweezers. Any ESD safe set.
Fume extractor: VERY important for health. You do NOT want to be breathing in flux fumes. A high-volume HIPAA air purifier on the desk works, ($150 or so) or a dedicated device like the Hakko FA430 is even better ($625).
Oscilloscope: Rigol DS1054-Z. 50MHz, hackable to 100MHz bandwidth easily. $400. There's no better cheap scope at the moment (IMO).
Function generator: Siglent SDG805. $270. Needed to give you analog signal inputs. Part of the big-3 of 'scope, power supply, and function gen.
Power supply: Get a linear supply. The Tekpower TP3005D-3 is $200, and is an actual linear power supply. The knobs are coarse adjust only (it's analog), I replaced the control potentiometers with 10-turn versions which substantially improved the accuracy of the output. There's also the Siglent SPD3303X-E ($340) if you want a digital panel version. You definitely need arbitrary +- voltages for lots of very basic circuits, PC power supplies are very limiting and too noisy if you do any sensitive analog design.
Multimeter: Get a safe one (HRC fuses, proper transient voltage suppression, etc.) Can't go wrong with Fluke, of course, but Extech, Brymen, and some others have cheap and capable handheld meters. $100-300, depending on brand. Be sure it has a micro-amp range! The really cheap ones don't, and you WILL need it if designing embedded stuff.
Logic Analyzer: Get a LogicPort. pctestinstruments.com. They're $390, for a 34-channel 500MHz device, very nice for the money. Needed if doing much digital work. (Keysight's 34-channel standalone analyzer is $12165 base price. 5GHz, but still, twelve grand...)
Spectrum Analyzer: If you're doing RF work (radio design), you'll need one. Otherwise skip it. The Siglent SSA3021X with tracking generator add-on is $1764 (pretty cheap) and quite capable (9kHz to 2GHz). It's also hackable / software upgradeable into the 3.5GHz model. The Rigol DSA815-TG is $1550, but significantly worse (smaller display, worse resolution bandwidth, max 1.5GHz, etc).
Be sure to get a GFCI outlet and a GFCI adapter or two. The oscilloscope, function gen, spectrum analyzer, etc, all are mains earth referenced, and should each have their own GFCI plug. If you accidentally connect the ground lead of any of them to something other than ground the GFCI will trip and prevent the ground traces from being blown up inside the device. They're about $20 each, well worth it IMO.
You might want an anti-static mat and wrist-strap.
Get a bunch of small drawers, eg https://www.amazon.com/gp/product/B000LDH3JC. Print labels for them, use them to store resistors, capacitors, and other types. You can fit two values of component in each drawer (though they don't come with enough dividers :/). You want at least 96 drawers for resistors and 32 for capacitors assuming you're buying 1% or 5% resistors and 10-20% capacitors (pretty normal). I bought a kit of resistors (https://www.amazon.com/gp/product/B017L9GKGK) and (https://www.amazon.com/Joe-Knows-Electronics-Value-Capacitor...) for capacitors (Joe Knows Electronics kits are good for stocking up, they have more of the most common components in their kits.)
Get some desoldering wick and a solder sucker too. Also some tip tinner, and/or a sal ammoniac block. Make sure you have a roll of kapton tape to hold parts down while you solder them (it survives high temperatures). If you'll be doing a lot of surface mount you'll want a reflow oven and solder paste.
EDIT: One tip I forgot, very important: When you buy parts (on DigiKey/Mouser or similar) make sure you buy extras. At least the number needed for the first volume discount or 10, whichever you can afford. 3x the number needed for the project at the minimum. You WILL drop parts, burn them out, and otherwise damage them. It's much easier if you already have spares, don't have to wait for shipping, and don't have to pay for shipping. This will also help you develop a parts library, as you do more projects you'll be likely to re-use common parts and already have many left over from past work.
Soldering iron: I like the Hakko FX-888D. $90-110 or so. They have better if you can afford it, but that one's very good. The FX-951 is the next step up, and can take micro-soldering handpieces and has the quick-change tips. It's about $240.
Get a chisel tip, eg Hakko T18-S3, a bevel tip (T18-S6), and a bent-conical tip (T18-BR02). The conical tip is perfect for lots of general purpose work, you can use the fine point or the sides of the bend. The back of the bend can be used for drag soldering, the inside of the bend makes soldering wires together easy. The chisel tip is good for soldering things with more thermal mass (PCB-mount heatsinks) and the bevel tip is pretty necessary for drag soldering on QFP and similar surface mount packages.
Hot air station: Probably something cheap from china, there aren't any particularly affordable name-brand ones that I know of. Weller has the WHA900 for around $600.
Magnification: Get at least one of the magnifying headsets ($8-10 on Amazon) and a desk magnifier with LED ring light. Better option is an AmScope stereo microscope, such as the SM-4NTP and a ring light for it like the LED-144W-ZK. $480 total.
PCB vise: I have an Aven 17010, it works pretty well. MUCH better than trying to hold a board in the helping hands.
Flux: Get liquid flux with a syringe. Amtech is the best, but there is a lot of counterfeit stuff out there, and Amtech doesn't sell it directly (bulk orders only). https://mailin.repair/amtech-nc-559-v2-30-cc-16160.html sells the real flux.
Tweezers. Any ESD safe set.
Fume extractor: VERY important for health. You do NOT want to be breathing in flux fumes. A high-volume HIPAA air purifier on the desk works, ($150 or so) or a dedicated device like the Hakko FA430 is even better ($625). Oscilloscope: Rigol DS1054-Z. 50MHz, hackable to 100MHz bandwidth easily. $400. There's no better cheap scope at the moment (IMO).
Function generator: Siglent SDG805. $270. Needed to give you analog signal inputs. Part of the big-3 of 'scope, power supply, and function gen.
Power supply: Get a linear supply. The Tekpower TP3005D-3 is $200, and is an actual linear power supply. The knobs are coarse adjust only (it's analog), I replaced the control potentiometers with 10-turn versions which substantially improved the accuracy of the output. There's also the Siglent SPD3303X-E ($340) if you want a digital panel version. You definitely need arbitrary +- voltages for lots of very basic circuits, PC power supplies are very limiting and too noisy if you do any sensitive analog design.
Multimeter: Get a safe one (HRC fuses, proper transient voltage suppression, etc.) Can't go wrong with Fluke, of course, but Extech, Brymen, and some others have cheap and capable handheld meters. $100-300, depending on brand. Be sure it has a micro-amp range! The really cheap ones don't, and you WILL need it if designing embedded stuff.
Logic Analyzer: Get a LogicPort. pctestinstruments.com. They're $390, for a 34-channel 500MHz device, very nice for the money. Needed if doing much digital work. (Keysight's 34-channel standalone analyzer is $12165 base price. 5GHz, but still, twelve grand...)
Spectrum Analyzer: If you're doing RF work (radio design), you'll need one. Otherwise skip it. The Siglent SSA3021X with tracking generator add-on is $1764 (pretty cheap) and quite capable (9kHz to 2GHz). It's also hackable / software upgradeable into the 3.5GHz model. The Rigol DSA815-TG is $1550, but significantly worse (smaller display, worse resolution bandwidth, max 1.5GHz, etc).
Be sure to get a GFCI outlet and a GFCI adapter or two. The oscilloscope, function gen, spectrum analyzer, etc, all are mains earth referenced, and should each have their own GFCI plug. If you accidentally connect the ground lead of any of them to something other than ground the GFCI will trip and prevent the ground traces from being blown up inside the device. They're about $20 each, well worth it IMO.
You might want an anti-static mat and wrist-strap.
Get a bunch of small drawers, eg https://www.amazon.com/gp/product/B000LDH3JC. Print labels for them, use them to store resistors, capacitors, and other types. You can fit two values of component in each drawer (though they don't come with enough dividers :/). You want at least 96 drawers for resistors and 32 for capacitors assuming you're buying 1% or 5% resistors and 10-20% capacitors (pretty normal). I bought a kit of resistors (https://www.amazon.com/gp/product/B017L9GKGK) and (https://www.amazon.com/Joe-Knows-Electronics-Value-Capacitor...) for capacitors (Joe Knows Electronics kits are good for stocking up, they have more of the most common components in their kits.)
Get some desoldering wick and a solder sucker too. Also some tip tinner, and/or a sal ammoniac block. Make sure you have a roll of kapton tape to hold parts down while you solder them (it survives high temperatures). If you'll be doing a lot of surface mount you'll want a reflow oven and solder paste.
EDIT: One tip I forgot, very important: When you buy parts (on DigiKey/Mouser or similar) make sure you buy extras. At least the number needed for the first volume discount or 10, whichever you can afford. 3x the number needed for the project at the minimum. You WILL drop parts, burn them out, and otherwise damage them. It's much easier if you already have spares, don't have to wait for shipping, and don't have to pay for shipping. This will also help you develop a parts library, as you do more projects you'll be likely to re-use common parts and already have many left over from past work.