To say "I wanted you to see (how big the world was!)" you would implement the subjunctive: volevo che tu vedessi. Composing sentences that deal with desire or opinion get more complicated...but here are some examples that may help clarify the role and placement of the participle:
- Noi abbiamo voluto mangiare la pizza/We have wanted to eat the pizza
- Noi abbiamo voluto mangiarla/We have wanted to eat it
- Noi l'abbiamo voluta mangiare/We have wanted it to eat it
With avere the participle does not change with gender and number except when the the direct object pronoun (in this case, what substitutes la pizza) attaches to the auxiliary.
As for understanding the sentence construction, observe the following:
- Io ti ho voluto vedere
- Io ho voluto vederti
- Io ho voluto vedere te
The third option would primarily be used to place emphasis on you, not exactly a construction you would use day to day, but hopefully the similarity of #3 to the English sentence "I have wanted to see you" clarifies the function of vedere. Later on, you'll learn how to express I wish that you... He hopes that you... We want you to... and most of the time, che will indicate that the sentence is not I want to see you but I want you to see.
Hope this helps :)