1) Tell your manager first in private. Talk only positively. Despite being there for such a long time this is something managers deal with on a frequent basis and expect. (Do not tell your coworkers anything until everything is official)
2) Be open to negotiation. You may have no idea how much they may be willing to put on the table or you may be able to up your offer at the new place through a series of counter offers. (If you have already accepted the new offer, only talk to your current employer)
3) Allow for 2 weeks notice starting at the beginning of next week. Be proactive in putting together a plan to transition your work over to other members of your team. (Also while relatively uncommon do not be surprised if they happen to ask you to leave immediately - its typically nothing personal but can be policy)
4) Get the personal contacts of your close colleagues. Spam everyone there with LinkedIn invites to connect. Possibly ask any close colleagues or supervisors for recommendations.
5) Collect and return any company equipment. Arrange for your last paycheck including any unused vacation.
6) On the last day, you will most likely take part in an exit interview. Say nothing negative - even constructive feedback should not be given. They will ask you to sign some paperwork. Tell them you will take it home with you to read in full. Do not send in the paperwork - it only is meant to clear your employer of any liability and void your legal rights.
On point 2, I'd like to offer a different view, matching my own experiences and those of people I know in other companies who give the same counsel. Once you've made it clear that you were about to leave and you stayed because they offered more money or some other such better deal, they will always, always, be wondering when you're going to actually leave, and that will affect how they plan your participation. It will change things. It's not that you won't be trusted, so much as you will have a track record of leaving. Unless the counter-offer really is spectacular, once you've decided to go, go.
> Also while relatively uncommon do not be surprised if they happen to ask you to leave immediately - its typically nothing personal but can be policy.
If this happens, they should pay you for the intervening time. IIRC, generally in the U.S., if they don't, then termination was initiated by them, not for cause, and you are entitled to unemployment insurance. (Though you may debate with yourself whether you want to go through the exercise. Nonetheless useful to keep in mind if your new job then falls through at the last moment.)
> They will ask you to sign some paperwork. Tell them you will take it home with you to read in full. Do not send in the paperwork - it only is meant to clear your employer of any liability and void your legal rights.
Don't sign anything without reading it thoroughly and understanding. Even if you are quite friendly with your manager, this is not coming from them, it is coming from Legal and HR. They are, most likely, not your friends -- not to sound overly spiteful; that's simply the way it works.
If they are offering you money to sign ("severance package", etc. -- not as likely when you are the one quitting), that is all the more reason / warning sign to be cautious.
Re point 2: As EliRivers has also stated, taking a counter offer when you have already stated your intent to leave can lead to a bad environment for both you and your employer. I don't see anything wrong with using that counter offer to try and negotiate a bit higher at the new job but personally I would suggest not taking counter offers.