Eventually, you probably want to get on to I-90 (the Mass Pike). This goes right into downtown Boston, so find a busy onramp downtown.
From South Station there is an on-ramp not too far away with a traffic light and space to pull over. The biggest problem is that there is limited traffic and probably most of it is local. But you should hopefully be able to get to the service plaza in Framingham from here. From the station cross the street (Atlantic Ave.) and keep heading away from it--west. You should go through China town. After a few blocks, turn left (it doesn't matter so much how far you go before you do). That should take you to Marginal Way, which parallels I-90. Turn right, and follow it until it turns into an on-ramp. That's the spot.
If you're in the suburbs already, a tempting spot to start might be by the Riverside MBTA station, but you'll probably find very little traffic going your way. If you can get a short ride just as far as one of the service plazas, you'll likely have much better luck.
If you're in Cambridge, a good spot is the on-ramp at Soldiers Field Rd and Cambridge St. Traffic is slow here and has space to stop on the shoulder.
Route 9 is an excellent road to hitchhike on: lots of traffic, plenty of stoplights and gas stations, and a decent average speed. You can probably get to Worcester faster along Route 9 than by taking the Mass Pike.
I-95 is your best bet. There is a service plaza just a block away from the Riverside MBTA station. Try to look as friendly and harmless as possible, as the area (Newton and Wellesly, MA) has a reputation for snobbish people who are likely to be quite cautious about picking up hitchhikers. Expect a medium to long wait in any event.
I-95 bypasses the city, so the main highway northeast is US-1. The onramp onto US-1 in Charlestown is fairly busy and sees some longer-distance traffic; the southbound side of Rutherford Ave near Community College MBTA station is a decent spot to hitch onto US-1 towards Maine. Out in the suburb of Saugus, US-1 stops being a full freeway and it may be feasible/legal to hitch directly along the road there (if you take a bus out that far). Further out than that, US-1 reaches a junction where most longer-distance traffic shifts to the I-95 freeway.
Buses are $2.00 (1.50 with a Charlie Card) including a transfer. Subways (the T) are $2.50 (2.00 with a Charlie Card) and likely faster. Use these to get yourself to a good hitchhiking spot or to your final destination once you're close.