Before you get involved with most plumbing projects, you'll need to shut off the water flowing to whatever you're working on. Most of the time, there are easy-to-access gate valves or compression valves that you can turn with your hand. Turn them clockwise all the way to turn off the water and counter-clockwise to turn it back on when you're done. For sinks, look under the sink and you'll usually see two valves—one for hot water and one for cold. On kitchen sinks, you might also see valves for the ice maker on your fridge or your dishwasher. Just turn them all off. For toilets, the valves are on the wall or right on the pipe behind the toilet.
Get a minimum of three bids. Estimates for an average-sized job should be within a few hundred dollars. Be suspicious of anything that is substantially lower or double the price of the rest, and watch out for hidden fees, like charges for travel expenses. They could be signs of a home improvement scam. A good plumber will not nickel and dime you like this, and many of us will offer free estimates.
I think it is rather rude and a bit dishonest for people to not divulge their total fees upfront, an of course we should know to ask. But the normal person wouldn't know the questions to ask until they are burnt once like this, and then there still could be costs for walking up stairs or charge another service call if they need to go somewhere to use the bathroom and come back. THAT ISN'T A QUESTION I WOULD THINK TO ASK.
Sometimes a faucet leak can be stopped by replacing the rubber washer at the end of a faucet stem or cartridge. This is a more common repair on older fixtures when things were built to last and designed to be able to be maintained. Other times the whole stem or cartridge needs replacement or the bibb seat deeper back into the fixture needs replacement. That leads to matching up the right components and often times making a trip to the hardware store or plumbing supply house necessary due to the many varieties, makes and models of faucets. My advice is to make sure your Plumber has faucet repair kit on their truck before having them come out to assess the issue.