You'll likely need to do some light plumbing work if the new sink drain is in a different location (meaning re-work the drain pipe and trap). Not too hard, you can get PVC parts at hardware store or big box store. Just do some reading on code, there can't be too many bends or pieces between the sink and the wall, and S-traps are not legal anymore. Other than that just reattach the water hot and cold supply lines according to the directions on your new faucet.

Since you're looking to do most of the labor yourself, your biggest expense is going to be the vanity followed by the counter top. It's crazy how much these two things will cost. For the vanity, consider checking out places that offer second hand building materials. Then, look into remnants the counter fabricator may have leftover from other jobs.

Master bathroom, wanting to change mirrors and lights, burn it wanting to spend a ton..: suggestions for biggest impact ? Want glam, modern / traditional

I wouldn’t do a lot until you have what it takes to redo this space. ( I would possibly swap the toilet and sink. ) For now, Paint the vanity. New countertops and if possibly hardware ( including faucets ) Paint the walls.