Hi, Denise ! If you have arthritis that makes it difficult to lift weights per se, you might consider opting for resistance bands instead. Resistance bands are a great way to strength train. If you're looking for some simple routines, join SERGEANTMAJORs resistance band team. He's got a bunch posted. www.sparkpeople.com/myspark/groups_individ
If you can't do push ups because of your wrists, then try wall push ups. Wall push ups are a perfectly good modification that can help you increase upper body strength.
The problem with many of today's popular strength training DVDs is that many do include some cardio as part of the workout. The one that doesn't might be too intense for you and that's Body Pump. Body Pump is a type of circuit training workout that uses barbells and hand weights. With your arthritis, it's going to be too intense.
One thing you might ask your doctor is whether or not you can do yoga. yoga has many poses that can help increase your strength. If you've never done it before, you should work with a good instructor. However, if you don't have the money to take a class, look for PEGGY CAPPY's DVDs. She teaches yoga poses with chairs for people who can't stand for long periods. Yoga may be a possible strength training option for you.
I know there are many who don't feel that yoga is strength training, but I disagree. Many poses do require quite a bit of strength to do them.