1たい form
Shows that you want to do (verb). You take the stem of a verb and add たい. Conjugates like an い adjective.
2Negative たい-Form
Take the たい form, drop the い, add くない
3Past たい-Form
Take the たい form, drop い, add かった
4Past, Negative たい-Form
Take the たい form, change to negative, add かった
5～たいと思っている
Shows that you have wanted to do (verb) for quite some time.
6たい-Form for Other People
You DO NOT usually use the たい form to show someone else's desires. You can use quotations, observations, or guesses instead. So using と言っている with たい would be okay.
7Verb in たい-Form + と言っている
"They said they wanted to (verb)"
8Verb Stem + たがる
"It seems that he/she/they wants to (verb)"
9Informal Past Tense Verb + り + Informal Past Tense Verb + りする
"Do things such as A and B"
You can use the て-form of a verb to say that you did A and B, but by using that sentence structure you make it seem like you ONLY did A and B and in that order. り/りする means that you did A and B, but there were some other things that you did as well.
The tense of the final する determines the overall tense of the sentence.
11Past tense sentence + ことがある
Saying that you've done something. The tense of ある dictates the tense of the whole sentence.
12Noun A + や + Noun B
"A and B" Acts like the と particle, except it's not exhaustive.