As to busing across the border instead of taking a taxi, it's easy. Generally you can just change to a Belize-bound bus (Novelo's, Northern Tranport or James) at the ADO terminal. At certain times, mainly early morning, you may need to take a short taxi ride to the Nuevo Mercado (New Market) station to catch the Belize bus -- ask in the ADO terminal or ask someone on the ADO bus, if it's not clear. There are buses from Chetumal heading to Corozal and then on to Belize City roughly every hour from early morning to late afternoon. The bus across the border is only a couple of bucks, and it's almost as fast as a taxi.
Unless you want to overnight in Corozal, which isn't a bad idea as it's such a pleasant place and a fraction of the price of San Pedro, you'll be better off flying from Coro to San Pedro. There are about five flights a day each on Maya Island and Tropic, less than US$40. Just ask the bus driver or attendant to drop you at the Corozal airstrip, just south of town.