As a software engineer, I've put my resignation in at a few jobs, and many more prior to starting my career. I've left some on great terms, and have left some on sour notes. As such, I figured I'd share a bit about what it means to quit with grace, and how you can leave behind a team while maintaining strong professional connections.
Notifying Your Manager
Firstly, and most importantly, is your manager. At the end of the day, this person will likely have the most weight in how the rest of your experience goes, and whether you end up burning bridges. I recommend that you set up a sit down meeting with them to let them know you've accepted another position. In that meeting, try your best to make it clear that it was purely business.
Remember, they are also a human being with feelings, and if they care about you and their job they may wonder if there was something they did or didn't do to cause you to leave. Let them know that you enjoyed reporting to them, and that their leadership, the company and the team as a whole were great. Of course, if there are complaints to lodge, you shouldn't blatantly lie. But you should be tactful and use a lot of discretion with which grievances you choose to voice, and how you word them. This isn't the time to complain about every small thing.
Notifying Your Team
After letting your manager know, you should let the rest of your team and close associates in on the change. I don't advise making some grandiose and dramatic farewell display. But you should make sure that everyone gets the memo, and then go out of your way to make any follow-up easy. You can do this by creating documentation for things you've worked on, setting up meetings to share your knowledge, and talking with coworkers directly to pass along information and projects that you're currently working on. Do your best to bring anything you are in the middle of to a clean tying off point.
Handling the Social Aspect
Along with leaving a company and team comes some typical (in the US) ceremonious socialization. It's important to set up the often awkward, uncomfortable last farewell event. Most of the time a simple lunch will suffice, but fit the event to whatever your team culture is. All you're looking for is the opportunity to see and say farewell to the people you've worked with. Along with that, it's also a nice gesture to consider getting a small farewell gift for any people who were especially beneficial or close at the company, for instance a mentor or team lead.
Bridging the Outer World Gap
The last step to gracefully depart is to proactively bridge the gap from office coworkers to professional connections. Find the people you interacted with often and add them on LinkedIn. Perhaps write them a short LinkedIn recommendation letter if they were really awesome, or provide them with a couple of endorsements for common skills you've worked together with. Who knows, they might return the favor.
The final, and perhaps most important piece of advice, is to not take things too seriously. After all, this is the end of a business arrangement, not a break up with a romantic partner. I can almost guarantee that none of your coworkers will shed tears for your departure. The day after you leave will likely just be another day in the office. So try not to overthink things, and try not to fool yourself into believing this is something more than it actually is. People are hired, people are fired, people quit for greener pastures. Most everyone you work with has been in the same place you are. Keep things professional, and you'll be off to a great start at your new company!