What are my options to change the finish on a faucet?
Lots of chatter in the comments, so here's my take that tries to take a few of those into account.

Paint/finish at home is probably a dead end. You can try, but I can't imagine how you'd get anything attractive at the outset, and I have even stronger doubts that it'd be durable.

Powder coating (really just a plastic paint that's baked on) tends to be cheaper, but not incredibly durable.

Electroplating is the way to go for a quality finish, but you have to weigh the cost/benefit of plating versus replacing. This faucet looks quite new from the picture, so it's not like it's at the end of its life cycle. Regulatory frameworks (and thus prices) vary a lot by jurisdiction, so shop around and be prepared to mail your faucet somewhere. Google 'electroplating' to find places. You might save a buck or two if you're prepared to take your faucet apart for them. They'll advise on what other preparation (ie, sandblasting to remove old varnish) is required for a quality finish.

If you want to dispose of the faucet because plating costs more than a new one, I'd encourage you to look for a place that re-uses building materials. Habitat for Humanity runs one called the ReStore.

My personal opinion is that I'd buy a new one in the finish I want.