By Metro
The main Metro stop for the Mall is the Smithsonian station on the Blue/Orange Line, which drops you out on the east end of the Mall, right by all the museums. The Smithsonian station is relatively close to the center of the park, so you won't have to walk much more than a mile to get to any sights although a mile on a hot summer day can be an unpleasant walk!. One stop south of Smithsonian on the Orange/Blue lines is Federal Center, which is very close to the southeast end of the Mall. If taking the Green/Yellow Line, Archives/Navy Memorial on the north and L'Enfant Plaza on the south are both very close to the museums.
West Potomac Park is rather poorly served by Metro. If you are going to the Lincoln Memorial, the closest station is actually across the border in Virginia at the Blue Line Arlington Cemetery station, just across the Arlington Memorial Bridge which has good views. The Blue/Orange Line Foggy Bottom station in the West End is a similar distance to the north. Either is a good ten minute walk from the Lincoln Memorial.
By bus
While Metro will connect you with most of the city without needing to take the bus, there are a couple key routes to be aware of especially to Metro-less Georgetown.
Routes #31 (http://wmata.com/bus/time...) and #32, #36 (http://wmata.com/bus/time...) all run northwest up Pennsylvania Ave to M St through Georgetown, and then up Wisconsin Ave all the way to Friendship Heights. You can catch them as far south as Independence Ave west of the Capitol Building after which they run up 7th St to Pennsylvania Ave.
Routes #52, #53, and #54 (http://wmata.com/bus/time...) provide a quick service up 14th St through the Mall to Logan Circle and U St.
The D.C. Circulator runs a very tourist-friendly service Sa-Su 10AM-6PM: the Smithsonian–National Gallery of Art "Purple" line, which rings the Mall from the Reflecting Pool in the west to the Capitol Building in the east. The Circulator's Convention Center–SW Waterfront "Red" line is a great way to move from the Mall along 7th St either up into Chinatown and the Convention Center, or down to the popular Waterfront and the Fish Wharf.
By taxi
Taxis are easy to hail all day and night, and can be a especially convenient option from Georgetown, Union Station, or National Airport. Aside from peak periods such as Cherry Blossom season, when traffic slows taxi travel to a halt, taxis can be a nice way to take a load off your feet and get from one end of the Mall to the other quickly.
By car
Driving in and around the Mall is a great recipe for a headache due to chronic traffic jams, unintuitive traffic patterns, and very limited parking. Driving towards a monument doesn't mean the road will lead you towards it more likely, it will hurl you across the river into Virginia. If there is any sort of special event going on, you can easily get stuck for hours. Parking garages throughout the West End and East End fill up early with office workers. Some may still have space available, but prices are steep at least $14 per day. Double check the garage hours, so you can exit before the garage closes!
On evenings and weekends during the winter, it is sometimes possible to find metered parking on the mall with two hour limits, although you should budget a lot of time to find a space just in case. There is no enforcement on Sundays or Federal holidays, so if you are lucky enough to find a spot, you can stay as long as you wish.
7th, 14th, and 17th St are the main north-south routes, while Pennsylvania and Independence Ave are the main routes for east-west traffic. If coming from Arlington, the simplest routes are I-395 to 14th St, or the Arlington Memorial Bridge to Independence Ave.