This is going to be a bit variable, but in simple terms:
1) If you match into a PGY1, you have to start it or you'll be permanently banned from the match. So you start the residency you matched. You can theoretically quit after a couple months, but that's a really dumb idea - you should start it and *work hard* so that you can get a good letter of rec from your program director
2) You then apply to the other specialty. You can apply the typical way (through the match) for PGY1 spots and if it's a specialty that will recognize your intern year, you can also apply to any PGY2 spots that start that coming year (R spots) through the match AND apply to any PGY2 spots outside the match that open up (some specialties have listservs for this, otherwise you literally just have to start searching). To be successful in this application, it's damn-near-required to have a letter from your current program director, even if all it says is "tinyhandsbob started residency two months ago and is a resident in good standing. They desire to change specialties due to feeling they would fit better with a different one".
Obviously you'd rather just start as a PGY2 if you can, but there's more PGY1 spots than PGY2 spots typically. Psych is weird, because there's so much elective time, they'll recognize just about any intern year and can just have you start as a PGY2 - still graduating on time, even if you did zero psych at all your first year. Now, they don't *have to*, but they often do.
3) Funding is a question we often get on this forum. More or less, you should just ignore it as a question. It almost always doesn't matter.
4) If you're an FMG, the only extra factor you'll have to consider is visa status. Because if you don't match and thus don't have a residency after your first year... you're very likely to fall out of status and end up being required to leave the US. It's the risk you take by giving up a sure thing (current residency) for potentially changing.