http://www.amazon.com/Conditioning-Climbers-Complete-Exercise-Series/dp/0762742283
btw this is the first and only climbing book i've purchased and in my personal opinion everything in this book is extremely beneficial as far as when it comes to proper conditioning and rest can be learned through trial and error through time put in on the wall.
If you look at many of the best climbers in the world (Daniel Woods, Chris Sharma, Paul Robinson, etc) they are all pretty skinny with powerful hands, arms, shoulders and abs. Being a good climber is all about reducing your body weight while strengthening specific muscle groups that help your climbing.
Weightlifting for climbing should focus on the benefit you want to get. You should go buy a book like this one to learn what you need to do to condition yourself for climbing.
Curing tendonitis is not a simple endeavor and lots of people here have offered good ideas that have worked for them. Unfortunately, current sports science and applied physiology have yet to really solve the great mystery behind how to cure tendon strain and most "cures" are anecdotal and are not founded in any peer reviewed science. Recent, understanding of tendon injuries suggest that taking time off may do nothing to help injuries go away and may lead to more injuries down the road (Brukner & Kahn's clinical sports medicine, 4th ed).
With that in mind, if you cannot afford a physical therapist or even just as good, a climbing coach, then your best bet is to become your own coach as Dave MacLeod says in his book "Make or Break", which I highly recommend adding to your library. You have to create your own formula that works for you, a combination of many of the ideas floating around this thread and more. Obviosuly, if paying for a PT is not an issue don't hesitate and find a qualified one ASAP.
I don't know much about how you train or how you climb but 9/10 times, injuries like what you've described are a sign to return to the basics. While, doing antagonistic training like the rice bucket, pronators, and pushup's might help the pain leave, you might find yourself reinjured in no time, like I did to myself. This the most imperative thing to solve and requires you to have a deep understanding for why you've injured yourself. Once these injuries surface, particularly tendon injuries, they become more and more likely to reoccur.
From my understanding, elbow injuries and shoulder injuries are often rooted in the weakness of quintessential stabilizing muscles in key places in your body: abs, shoulder, back. Intensive and highly specific movements found in climbing require not just merely okay stabilizers but strong and robust ones. Focusing your training and climbing with just powerful movements like big moves and pull-ups will cause your body to compensate in ways that puts more stress and strain on joints and tendons.
If you're interested in becoming your own coach I recommend these books for starters as they did a lot to aid me:
Take my words with a grain of salt. I have no degrees in any medicine, physiology, or sports fields. I'm simply a climber who has had these injuries plague me for over a two years. I have read everything from dissertations to the listed books above. Furthermore, I've been able to afford professional physical therapy for the past 2+ months and have verified a lot of the information I've read with them. There is a lot of misinformation out there on the internet on this subject so be skeptical.