"lavarse" and "acostarse" are reflexive verbs in Spanish, which means that the subject performs the action on him/herself ("-se" can be translated as "oneself").
You cannot use "lavarse" when you're talking about washing dogs or other things because it means "to wash oneself". If you take off the reflexive "se", it just means "to wash": Voy a lavar la perra por la mañana".
When you conjugate the verb, you have to change the reflexive pronoun "se" according to the person of the subject: me lavo, te lavas, se lava, nos lavamos, os laváis, se lavan (I wash myself, you wash yourself, he/she washes him/herself, etc.) (the reflexive pronoun generally moves in front of the verb as well, except in the infinitive, gerund and imperative), so "you will go to bed" will be "vas a acostarte" ("I will go to bed" > "voy a acostarme", etc.).
In No. 2, "comprar" is correct, but you put in an extra "ir" ("to go"). "Va" ("she goes") is part of the verb "ir" (which is irregular): "Ir a comprar" = "To go to buy", "va a comprar" = "she goes to buy (she will buy)".