La Plata is the capital of Buenos Aires province in Argentina.
North towards Buenos Aires
There's a roundabout followed by a series of speed bumps at the beginning of the highway leaving La Plata. You can take a number of buses (for sure the 307 A and E, but others, too) to the roundabout. At the first speed bump, you have good visibility as cars enter from the roundabout, and there's space for them to pull over just past the speed bump.
If you can't get a ride and want to take a bus, buses pass rather frequently, and you can catch one on the opposite side of the roundabout. With a Sube pass, it costs 25 pesos (as of April 2014).
Alternatively, the train to Buenos Aires costs less than a local bus within La Plata (2.80 pesos vs 3 pesos as of April 2014).
South towards Mar del Plata
To go to Mar del Plata you need to take Ruta 2, which passes close to La Plata. You can take a bus 14 from the city centre (the last stop is at the intersection of Calle 1 and Calle 57) to Olmos. It costs 4 peso (June 2015) and you need a subte card to pay, but it is not a problem to give cash to someone and ask to pay by their card for you. Ask a driver to leave you at the traffic light after Olmos, where a bus turns off from the main road. From there you have 3-4 km untill Ruta 2, it is easy to hitch a short ride there. Once you get to the higway, walk 1 km to the south, where a YPF gas station is. It is a good spot to hitch, where you can get a direct ride to Mar del Plata.
Other useful info
If you buy a Sube pass to ride the buses in La Plata, it's the same pass for the buses and subte in Buenos Aires.