This.
The key thing is separating out the strands of what isn’t working for you, where specifically (if anywhere) those things would be different, and what you would give up to change it (because there are costs, always, to a move).
So:
What exactly is making you ‘extremely unhappy’? being far from home? adapting to US culture? some specific aspect of campus culture at Princeton? the academics being only ‘satisfactory’? something you expected to be true that isn’t? getting less than perfect grades for the first time*? Which of those things will change if you move schools?
Where would those things be different? Have you gone to visit any of the places you are considering yet*? If not, get your skates on and do the information tours at each of the places you are seriously considering (you might skip Stanford, as it will be hard to fit in on a weekend, but the others can all be done by train as a day trip from Princeton). For reference, do the standard information tour at Princeton as well, because you can compare the difference between how it is presented and your experience of the reality- all info tours are at heart sales and marketing tools.
And finally: what are the costs of change (on both ends- what you leave, and what you move to)? What relationships (with peers, profs) have you started to form? Columbia has a big set of ‘core’ requirements, which has several layers of implications- have you thought those through? You will be the new student, again, only this time most people really do have something of a friend group (even though there is still fluidity).
By all means, go ahead and do your homework, and file some transfer applications. If you only want to move if you can stay at the same level, leave it there, but if you definitely absolutely have to move, include a few places where the acceptance rate is likely to be higher.
*I know you did a summer program at Yale, but be careful about that memory: comparing a summer program for secondary school students to everyday life for university students is apples and oranges- still fruit, but with significant, meaningful, differences.
**As others have posted, your GPA is actually fine for a first term at uni, esp if it ends up at the higher end. Remember that GPA has no intrinsic value beyond applying to grad school, and for STEM grad school GPA is not the critical factor. A 3.6 - 3.8 GPA from Princeton will leave you qualified fo any grad school program in the world. What will matter in applications are 1) your practical experience; 2) LoRs and 3) for some subjects in the US, GRE scores. If you don’t have a summer research job lined up yet, that is urgent- deadlines are upon you. Start in your own department- they will have lots of info on options (both on campus and at other institutions). Fwiw, on-campus internships are often the best bet for 1st years (even if you transfer in Sept you can get a summer of experience!).