1) à ≠ de

These eight verbs have different meanings depending on which preposition follows.

*décider qqun à* to persuade, convince décider de to decide (to do)
*demander à qqun* to ask (for permission) demander de to ask (to do)
*jouer à* to play a game or sport jouer de to play an instrument
*manquer à qqun* to miss (someone) manquer de to neglect (to do)
*parler à qqun* to talk to parler de to talk about
*penser à* to think about (imagine) penser de to think about (opinion)
*profiter à* to benefit profiter de to make the most of
*venir à* to happen to venir de to have just (done)

Par exemple…

J’ai décidé de voyager. I decided to travel.
J’ai décidé ma sœur à me rejoindre. I convinced my sister to join me.

2) à + de

Nine verbs can use à and de together in the following construction:

verb + à + quelqu’un + de + infinitive

For eight of these verbs, à indicates who is to do something, while de precedes whatever that something is.

à qqun de faire
commander to order someone to do something
conseiller to advise " "
défendre to forbid " "
demander to ask " "
dire to tell " "
interdire to forbid " "
ordonner to order " "
permettre to allow " "

So for the above verbs, the "someone" after à (or replaced by an indirect object pronoun) is the person who is supposed to perform the action after de.

Il a conseillé à Ana de partir immédiatement. He advised Ana to leave immediately.
Je te dis de manger les épinards. I’m telling you to eat the spinach.

The ninth verb, promettre, is a little different: the person doing the action after de is the same as the subject. À precedes the person who is promised that action.

Elle a promis à Paul de dire la vérité. She promised Paul that she’d tell the truth.
Je te promets de finir mes devoirs. I promise you I’ll finish my homework.

3) à = de

After these verbs, either preposition can be used with no real difference in meaning.

*commencer à/ commencer de* to start (doing)
*continuer à/ continuer de* to continue (doing)
*prendre garde à/ prendre garde de* to be careful to
*rêver à/ rêver de* to dream of, about

À and de in phrases

A few phrases look similar but have different meanings depending on the preposition.

à côté nearby, next to de côté sideways
à côté de next to, beside du côté de from (direction)
à la hauteur at the level [un mètre] de hauteur [one meter] tall