None of the the buses stay exclusively within the free travel zone. You can get on any bus, but all of them are just passing through the downtown area and will continue to travel on to a destination outside the free zone, so you will need to know where the boundaries are. Here is a link to the map that you can print out and take with you: http://metro.kingcounty.gov/tops/bus/area_maps/m_seattle-rfa.html
These are roughly the boundaries:
South Boundary: Jackson Street
North Boundary: 6th Ave (with a slight bump north along Olive Way to 8th Ave.) and Battery Street to the Northwest
East Boundary - the I-5 Freeway
West Boundary - Puget Sound
Every bus stop inside the downtown core area also has a map showing the boundaries of the the free ride area. Another option is to use the bus tunnel. The buses (but not the light rail which also use the tunnel) are free between all of the bus tunnel stops with the exception of the International district stop - the last one to the south. You will also find that the downtown area is actually very walkable with most things you will want to see within a half mile of Westlake Center. For example the Pike Place market is only about 4 blocks from Westlake Center. Here's a map and info: http://www.squidoo.com/Seattle-Pike-Place-Market
The Seattle Center (where the Space Needle, http://www.squidoo.com/seattle-space-needle, EMP and Pacific Science Center are located) is just a few blocks outside the ride free zone, so you can get off inside the boundary and walk the 4 blocks or so to the Seattle Center. Ask the bus driver to let you know when it is the the last stop inside the ride free area. Most will announce it anyway, but you can ask them to let you know to be on the safe side.
Another option is to take the monorail from the upper floor of the Westlake Center to the Seattle Center. It costs $2 each way, but makes the trip fast, fun and easy: http://www.squidoo.com/seattle-monorail
Another thing to remember is that the free bus area is not free after 7pm, so plan accordingly. You can also buy a Orca pass, which can be used for getting around on buses, light rail or Washington State Ferries easy and pretty inexpensive: http://metro.kingcounty.gov/tops/bus/fare/tickets.html
I hope this info helps.