I have lots of little bumps on my arms and back of my legs (these have always been there) and lots of dry patches. What do you think it is and what should I do about it?

Dermatologist- asap. since you did not upload a photo of these skin areas,they must be visualized to make any kind of diagnosis. Re-submit your question with a photo attachment. For plain dry skin I advise Nivea or LubriDerm.
Bumps on arms/ legs. https://www.aad.org/public/diseases/bumps-and-growths/keratosis-pilaris It is associated with dry skin moisterizing helps. I have found a shower scrub of sugar and coconut oil helps as well.
Dermatitis. It is difficult to comment without having a look at the bumps and dry patches but this could be dermatitis. Please visit a family doctor who can examine properly and then make a diagnosis. In the meantime try and use simple moisturisers to keep the skin moist at all times.
Rash. Two top choices would be eczema and or psoriasis. Given you have had for awhile, I would get checked. You could send pic to one of us! Easy consult!
Atopic Dermatitis. Its a type of skin problem that is seasonal and resolves with topical medication. Lac-Hydrin lotion applied may help. You can get that at the pharmacy now, used to be on prescription.