Learn Spanish (Latin America): ¿Puedes lavar el carro, por favor?

Just like with the tener que structure you are learning, when two verbs work together as a single idea, only the first one is conjugated. Here, puedes (you can) is conjugated, so the second verb, lavar (to wash), stays in its original infinitive form.

Do I need a word like que between puedes and lavar, the way I do with tener que?

No, you don't! The phrase tener que (to have to) is a special set phrase that always needs que. With most other verbs, including poder (can / to be able to), you just attach the infinitive directly after it: puedes lavar (you can wash).

Why does puedes have a "ue" in it if the base verb is poder?

Poder is a "stem-changing" verb. For most subjects (like yo, tú, and él/ella), the "o" in the middle of the word changes to "ue" when conjugated. That is why it becomes tú puedes instead of "tú podes".

I thought the word for "car" was coche or auto. Is carro correct?

Yes! Carro is the most common word for "car" in many parts of Latin America, such as Mexico and Colombia. You will also hear auto in places like Argentina and Chile. Coche is primarily used in Spain, though people will still understand you if you use it in Latin America.