烏天狗 said:
I thought that expressing "want" in the context of wanting to do something
If there's one piece of advice I'd give you, it's to try to break the habit of thinking of Japanese as a having 1-to-1 (or even a 1-to-0.999) correlation with English as soon as you can.
What I mean by this is: just because the ~たい form might be used to express "want" in the sense of expressing one's own desire, it doesn't mean it corresponds to all cases where we use "want" in English.
When you think about it, an expression like "Do you want to go see a movie on Saturday?" isn't really asking the other person "Do you (currently) have a desire to see a movie on Saturday?" at all. On the contrary, the idea of seeing a movie might not have even occurred to the other person until you mentioned it. Beneath the surface, it's actually just a way of extending an invitation, and we just happen to use "want" to express this in English.
In Japanese, this is expressed in other ways (as Majestic and mdchachi have explained in detail). In addition to the phrases they gave you, I'd also like to point out that one of the most common ways to extend an invitation is with the negative verb form.
今度の土曜日、映画(を)観に行きませんか?(polite です/ます form)
今度の土曜日、映画(を)観に行かない?(casual plain form)
(*I put the を in parentheses since it's often dropped in colloquial speech.)
You can think of this as equivalent to "Won't you go see a movie (with me)?" if it helps you make the connection, but in general, you shouldn't try to force Japanese grammar to conform to your understanding of how things work in English. I would say that an important hurdle to get over if you want to progress from the beginner to intermediate level is to not think in terms of "How do I say 'want' (or 'would', or 'can') in Japanese?", but rather "I'm trying to express [this sentiment] in [this situation]. How do I do this in (natural) Japanese?"
As an illustration of this, "I can speak Japanese." and "Can I borrow a pen?" both contain the word "can" in English, but they would be expressed in Japanese using completely different structures. You have to be able to process it not as "How do I express 'can' in Japanese?" but rather in terms of "expressing ability" and "asking for permission", and learning the appropriate Japanese phrases for each.
I realize that my commentary has gotten a bit philosophical here, but I believe this is an important concept to understand. (It's definitely one my Japanese teachers impressed upon me back in the day, and one I always tried to help my students understand back when I was a grad student teaching the language.)
At any rate, questions on this and other topics are always welcome here. Good luck with your studies!