We are visiting Lisbon and The Algarve (in July) and really do not want the responsibility of a car. Is this area doable by train?
Lisbon is very doable without a car and actually preferable without one, as parking and traffic are a pain. However, the Algarve wouldn’t work at all iMO without one. The bus service is sketchy and infrequent and the trains only get you from town to town and not within the towns.
The good news is that driving in the Algarve is easy and parking is not a huge deal.
Where on the Algarve are you staying?
You can get by train from Lisbon to the larger Algarvian towns. Having a car whilst on the Algarve is useful if you want to explore the area. Personally, I would never visit the Algarve without hiring a car, which is easy to do.
I'm not sure why the bus service is described as "sketchy." Whether or not it's "doable" may depend on the specific destinations you want to visit. Rick's guidebook summarizes transport connections for any destinations described there. You can also see www.algarvebus.info, www.eva-bus.com, and rail schedules at www.cp.pt.