General Strategies for Tendonitis
The real general advice is something you probably already know. Listen to your body, and if something hurts, quit doing it. If it hurts when you're not active, take a few days off. And if it continues to hurt after a week or more of rest, go see a doctor and physical therapist.
Advice for gym climbing
It sounds like you're coming straight "off the couch" and trying to do indoor bouldering as much as you can, as hard as you can. I'd recommend caution about that. That sounds like a receipt for tendon injury, exactly what you're trying to avoid.
Instead, just ramp up slowly. All of the structured climbing training plans I've ever seen all assumed that the athlete was experienced and well conditioned, and in pretty good climbing-shape to begin with. If you're not up to a basic level of climbing conditioning, the aggressive kind of training I think you're trying to do will just injure you.
Ramp up more slowly, focus on technique, having fun
If you want to get back into the gym and you're worried about overdoing it, don't try to train for multiple sessions a day. Just warm up and do one session, then go home. Don't try to climb more than 2 or 3 times a week, and not on consecutive days.
When you're in the gym on a given night, warm up and stretch first. Then climb whatever easy boulder problems you can. Focus on things you can easily onsite. Gradually work through problems you've done before, until you're repeating routes just below your limit. Then try and project one or two problems at your limit.
Its time to go home for the day when:
- You find yourself repeatedly falling out of moves you could do earlier in the night
- You find yourself having to crimp on every single hold, including ones you can usually grab with an open-hand grip. More than anything else, this is a sign that you've exhausted your fingers, and that your risk of tendon injury is starting to rise.
You can usually pick up lots of technical pointers by watching other people climb the same routes as you, and just hanging out in the bouldering area. You could also look into a skills class or group coaching class, if your gym offers one.
"Novice Gains" are your friend
This kind of very basic, semi-structured bouldering will still wear you out, and produce a positive training affect. You'll develop more strength and endurance in your fingers, build a better base of technique, and get to know more climbers in the gym all without having to kill yourself trying to train multiple sessions a day, or being on a super strict training plan. Take advantage of this. Ride that pony until it dies. Then go out looking for a more aggressive plan.