Going somewhere to do something with 'andare', 'venire' and 'tornare a' + infinitive
When you go somewhere to do something, Italian uses a fixed pattern: a verb of motion, then the little preposition a, then the plain infinitive of the activity. Vado a mangiare means I'm going to eat, and andiamo a ballare means we're going dancing. The same pattern works with venire (to come) and tornare (to go back). It also sits inside many everyday expressions, such as andare a trovare (to visit) and passare a prendere (to pick someone up).
English speakers trip over this pattern because English says the same idea in several different ways: go dancing, go and eat, go to eat, come see me. Italian ignores all of these and always uses a + infinitive. It never uses per, never a form ending in -ando, and never e (and).
The first verb shows the movement and changes for the person and tense. The second verb stays in the infinitive, the dictionary form ending in -are, -ere or -ire. The a links them. Any activity can go in the second slot: ballare, mangiare, studiare, dormire.
Rule of thumb: if someone physically moves in order to do something, put a between the motion verb and the infinitive. Where the second verb starts with a vowel, many speakers write ad instead of a, as in vado ad aprire.
Andare is the most common. Venire is used for movement towards the speaker or the listener. Tornare is used for going back somewhere to do something.
Some combinations are so common that they work like single words. Andare a dormire means to go to bed. Andare a fare shopping and andare a fare acquisti both mean to go shopping.
To visit a person is andare a trovare, or venire a trovare when the visit is to the speaker. Only use these for people. To visit a museum or a city, use visitare.
Andare a prendere means going to fetch something or someone. Passare a prendere means stopping by to pick someone up, usually by car. Venire a prendere means coming to pick someone up.
English often uses going to to talk about plans or the future, even when nobody goes anywhere. Italian does not use andare a for that. It uses the present tense or the future tense.
With a pronoun such as la or mi, you can attach it to the infinitive (vado a trovarla) or put it before the motion verb (la vado a trovare). Both are correct.
Tornare a + infinitive can also mean doing something again, with no movement at all. For example, torna a piovere means it's raining again.
The typical mistake is translating the English word for word, as in *vado per mangiare or *vado e mangio. Always use a + infinitive instead.
When someone goes, comes or goes back somewhere to do something, use andare, venire or tornare + a + infinitive: vado a dormire, vieni a trovarmi, torno a lavorare. Never use per or e here. For plans that involve no movement, simply use the present or future tense.