The Morogoro region, while not as rocky as other areas in Tanzania, still offers quite a few climbing options.
If you want to climb in the Morogoro area, you must ask for permission from the people who own the property. Additionally, you should always bring a machete (Swahili: panga) to clear away vines that may be obscuring routes.
In addition to the areas mentioned here, there are many nice boulders in the mountains if you take a hike to Morningside, or Lupanga peak, or any of the other local hikes. If you take a hike into the mountains, it could make a pretty good day. It is not advisable to hike into the mountains without a guide, or at least someone fluent in Swahili who knows where they are going and can ask for permission.
To reach this area, there are busses that you can take from Dar es Salaam, or Moshi.