On foot
You can of course walk to Chinatown. If coming from Brooklyn, you can cross the Manhattan Bridge, which opened to pedestrian traffic within the past few years. Note though, that your view will be partially blocked by a protective mesh, and that you will be periodically rattled by the B, D, N, and Q trains crossing the bridge. But on the plus side, you will exit on Bowery near Canal St. in central Chinatown. Another bridge that can be crossed from Brooklyn to Manhattan is the Williamsburg Bridge. After crossing the Williamsburg, you will be left at the border of Chinatown on Delancey Street. All things being equal, though, it is most pleasant to cross the Brooklyn Bridge and then take the short walk to the southern reaches of Chinatown from the pedestrian exit. Note that it is also possible to use a bike path on the Manhattan Bridge and that the walkway over the Brooklyn Bridge doubles for most of its length as a bike path.
By Long Distance Bus
Chinatown is the home of several super-cheap long distance bus companies, with Fung Wah Bus (http://www.fungwahbus.com/) being the granddaddy of them all. You can take buses from Manhattan's Chinatown to other Chinatowns in Boston; Philadelphia; Washington, D.C.; to various cities further afield; and to casinos in Atlantic City. Ticket offices of the various bus companies are scattered throughout Chinatown; besides Fung Wah Bus, there's also Eastern Coach (https://www.nydcexpress.com/), Lucky Star Bus (http://www.luckystarbus.com/), New Century Bus (http://www.2000coach.com/), and Sky Express Bus (http://www.skyexpressbus.com/).
By Subway
For general purposes, the D or B subway lines to Grand St. are optimal for accessing Chinatown. The J and Z to Bowery leave you a little north of the center of Chinatown. The F train to East Broadway leaves you toward the eastern edge of the neighborhood. The 6, R, N, Q, J or Z to Canal St. leave you a few blocks west of the center of the neighborhood though in the midst of the excitement, congestion, and vendors of Canal St this is generally the best stop for shopping for anything other than foodstuffs. Further afield, it is also possible to take the 4 or 5 to Brooklyn Bridge or the 2 or 3 to Park Place and walk north and east. The A, C, or E trains that stop at Canal and 6th Av. and the 1 train, which stops at Canal and Varick, are far west of the neighborhood though walkable in good weather.