Just take Buquebus. There are a couple options, but the most popular is definitely the way to go. You can take either a fast ferry (50 minutes) or a slow boat (3 hours); the fast boat costs a bit more. Both arrive at Colonia del Sacramento, Uruguay.
We booked e-tickets for the fast ferry one day in advance. Day of, we took a cab to the modern--though still under construction--ferry terminal in Puerto Madero. Once there, we just handed over our printed out receipt and that was that. (If you've got big luggage, you'll have to check it, as on a plane.)
Next, we walked upstairs where an Argentine immigration agent checked us out and a Uruguayan immigration official checked us in at the same desk. No trouble there.
On board the ferry, there's a small bar, but we shoulda brought our own snacks and drinks. Repurposed airline seats are bolted to the floor, but they're far more comfortable on the ferry than when strapped inside a metal tube at 35,000 feet. We didn't see first class but it probably wasn't worth it.
On arrival in Colonia we breezed through customs then walked out of the terminal into Uruguay. After a short walk through a construction area--what promises to be a new terminal--we hailed a cab and cruised into town. A ride to the center is US$4. (There's no ATM on arrival, but luckily we had some US singles to pay our taxi driver with.)
Of course you can also fly. Montevideo's international airport is on the fringes of town. Most flights come in from Buenos Aires, Chile and Brazil. Be ready for a long day of travel if you're coming from the States or Europe.