We submitted our paper and got rejected. Now we want to submit somewhere else. Me (first author) wants to submit the paper as it is, while the corresponding author wants to do further experiments. Is there anything I can do? Who has the last word? Can the corresponding just decide like that, even if the first author disagrees? And would it help me to post the work on a preprint server, and in that case, would I need approval from every co-author to do so, or does the corresponding decide?
Generally every author needs to agree on a publication; even on a preprint server, as long as their name is on it. Nobody should publish something with your name against your will (which is a good thing), and neither can you publish against the will of the others.
So unfortunately if they insist on doing further work, there is nothing you can do apart from trying to convince them that this is not needed, and that you need this published quickly.
Note however that scientific papers are there in the first place to progress our knowledge, not to make up people's CVs. Your co-author may have good scientific reasons for why they want to do further experiments, and it may improve the quality of the paper (of course based on the given information we here cannot have an opinion on whether this is so). This should be a very good reason to do the additional experiments, and it may well increase the chance of acceptance, which then would also be in your own interest.
I think the most important thing here is communication - you need to tell them your reasons but you also need to understand theirs. They may change their mind, or you may accept that there is some sense in doing the additional work.
Two notes to add to the answer of Christian Hennig. First it is generally not a good career move to fight with your supervisor (and another PI) whose recommendations you need about as much as you need any single publication. Don't prove yourself a poor collaborator early in your career.
Second, the time to publication of a submission "as is" elsewhere isn't necessarily shorter than doing what the journal suggested and resubmitting. It might even be longer.