Your question about whether to remove red sprouts at the base and the top of a rose bush begs more information.  A picture may help me understand your question.  Also, what kind of rose do you have? The answer could depend on knowing that.

A rose like any other perennial will grow new branches or canes throughout its life.  When one first emerges it may be red, only to turn green or brown later.  It your rose is a grafted rose, you will want to remove any new branches that grow below the graft.  If it is not grafted, you may want to selectively remove sucker growth emerging from the ground.

Red sprouts at the top of the bush are likely new branches that will produce flowers. Roses are pruned to remove old, diseased wood so that new branches will be formed and to shape the bush to be a healthy plant.

Oregon State University Extension has a publication called "Healthy Roses" with a wealth of information on roses, including how to prune them and how to treat sucker growth.  I hope it answers your questions.  

http://extension.oregonstate.edu/lane/node/147

If not, please ask us again including a photo to show the sprouts and the rose bush so we can give you a more helpful answer.