I am surprised by the amount of selfish responses. To me it comes down to professionalism and not treating the people I work with, with disrespect.
1. Find out your company's policy. For a lot of companies in the US it's 2 weeks, for a lot of Indian based companies is 3 months. So figure out how long ahead of time you need to let them know. (let's call this X)
2. Figure out how much PTO you have. (let's call this Y)
3. Do you have a new job lined up? When is their start date? (let's call this Z)
4. Ideally you want X + Y < Z, so you can take your PTO and work however many weeks you have and then start your new job. Otherwise you can X < Z + Y (where you are taking PTO and working on your new job)
Personally I don't like taking an extra break between work, so I have my last day on a Friday and then on Monday I am starting the new job. If you have PTO time you can at least get paid for not working. I have had about 11 different jobs and I can still contact old co-workers and managers and get recommendations when I need them. No reason to burn bridges.
Even in very toxic environments there are still some co-workers that I appreciate, or as a manager people that worked for me that I appreciated.
Having said all of that to respond to your question
> Do people generally just abandon the work in progress that they're doing or do they finish it out?
There is always a work in progress. Isn't that what working is? This is why you give your 2 weeks notice or whatever. If it's not time sensitive you can talk to your manager, tell them that you are going to quit, but work with him on the exact date. What you do in that 2 week time frame is either finishing what you want finish OR documenting / training the next person to take over.
One of the personal things I do, and I am probably a rare case, is to give everyone that mattered me to a hand written note of what I like about them, anything I am thankful for etc. With my direct manager I do the same, but I will add what I feel was the most frustrating point of working for them. There is no menace there, but more like a "hey I am leaving, here are my honest thoughts". I tend to do this the last day, drop things on people's desks, preferably when they are not around.
> Other people seem to move jobs a lot more than me.
I was that person, I moved jobs, a new job every 2-4 years. Any of the 4 year spans usually involved some internal move. Now that I am older, and perhaps wiser I have realized that if you find a company you like, it's fine to stick with it and grow with them. I have been with my current company for 8 years and I haven't really thought about leaving. I have changed 4 titles though, Jr Dev to Dev Lead with everything in between.