I replaced my bathroom lavatory faucet after my first outing. There are a few other threads on this sight that deal with this subject. Most people find the spout too short on the factory faucet; we tend to get water all over the countertop just trying to rinse our hands off. I went with a single handle, brushed nickel faucet from Menard's. It was their cheap store brand, but hey, it's better than the one that came in the trailer.
It's pretty tight to operate inside the cabinet, so I'd recommend getting a "basin wrench" - $8 - $12 at any hardware store. As others have said, don't remove the sink!
Just follow the water lines up with your hand while guiding the basin wrench into place. Remove the water line nuts and spud nuts (the large nuts that hold the faucet to the sink) from the old faucet to remove it. I found that my sink is such thin plastic, that there wasn't enough thread on the new faucet valves to tighten the faucet down to the sink. I just picked up 4 fender washers to use as spacers (two on each side), and that allowed me to tighten the spud nuts for a nice tight fit. If the water lines on your new faucet go through the center hole of the sink, the faucet will come with a tie-down bracket, so you may not encounter this issue.
Just a couple of other points:
1) My new faucet's spout is so long (or rather, the sink is so small), that now I get water all over the countertop in front of the sink, rather than behind it! Oh well!
2) Because the tail piece is glued to the sink, you won't be able to remove it and use the tail piece/pop-up drain that comes with your new faucet. You may be able to find a hole plug to fill the hole in the faucet where the pop-up stem is suppose to go. If not, a dab of silicone sealant will prevent water from dripping down through that hole.
3) If you have an acrylic sink or your sink is mounted underneath your granite or Corian countertop, do not use plumber's putty to seal around the bottom of the faucet. Plumber's putty will stain all three types of products.
Instead, use clear silicone sealant, or the vinyl flange that comes with most modern faucets.
It's really quite an easy job, but don't tell the DW! Stay in the trailer for a couple of hours with a cold one watching the game. Let her think it took you all afternoon, but you were glad to do it to make her happy!