Do I need to cite an old theorem, if I've strengthened it, wrote my own theorem statement, with a different proof?

I've done everything in my research completely from scratch. I require no outside results to prove my theorems (I'm writing a blog about this, mostly because I don't have time to write papers; here is a link to the blog). My methods are new and almost unrecognizable if you're coming from the classical theory. But obviously I didn't come up with these things in a vacuum, and I searched and there have been others with very similar ideas. I read lots of stuff and synthesized it into new stuff that does the same thing but a little bit better. This is what every researcher does to one extent or another. Would you even have thought of this if you hadn't seen the past result? Probably not. So give them credit!

Adding this only because I think the point wasn't emphasized enough in the other answers, but it may actually hurt the credibility - or, at least, the perceived professionalism - of your paper if you did not cite the old result. An important part of research is to establish context and relevance, and anyone who happens to know or find the old result might (wrongly) conclude that you didn't carry out that part as expected if you didn't cite the prior work.

I'm not 100% sure, but you should always cite something if you can. If it inspired your result, you should be citing it.