Could a US President decide they want to step down and arrange for someone else of their choosing to take their place?
Yes, it is possible. Although mostly theoretical.
This requires the following steps:
Convince the VP to step down voluntarily - the President can't fire the VP -, or remove the VP from office in some other way.
Nominate your pick for VP and get it confirmed by Congress.
However, the actions taken to achieve step 1 are likely to influence the confirmation process in step 2. So I doubt it can be pulled off in practice.
There is some historical precedent worth mentioning: Nixon nominated Ford for VP after his original VP had to resign. When Nixon himself resigned due to the Watergate scandal eight months later, Ford succeeded Nixon.
Note that Nixon hadn't planned to resign at that time yet, so it doesn't exactly match your scenario.
The only way to achieve that would be to amend the constitution, which requires the cooperation of 2/3 of each house of congress and of 3/4 of the states.
The order of succession is specified by the 25th amendment.
In other words, it is so impractical as to be virtually impossible.