Your best option will be the subway, but you should allow about 30-35 minutes total for the trip. You will still have to do some walking. Walking the entire trip will probably take an hour or more, and an Uber or taxi could take quite a while depending on the time of day, and will certainly be more expensive.
The simplest route would be taking the C train from 34th St-Penn Station to the 81st Street- Museum of Natural History Station; then you cross Central Park to the Met. That involves more walking than other options, but doesn't require that you change trains.
Alternatively, you could walk to 34th St-Hudson Yards and take the 7 train to Grand Central Station, and then catch the 4/5/6 train to 86th St. Then you walk a few blocks over to the Met.
If you have Google Maps on your phone, it provides subway directions.
One ride on the subway is $2.75; you do not have to pay again when you change trains. there is a $1 charge for a new Metro Card, and you also get a bonus added to your card if you put at least $5.50 on it. Several people can share a Metro Card, as long as it isn't the unlimited card. We always just get one for our group.