All three verbs can generally be translated to “to change”. Their meaning is quite similar. However, there’s a slight difference. Take a look:

1. etwas austauschen = (to) exchange sth./ (to) substitute

Example: Ich tausche meine Erfahrungen mit dir aus. I exchange my experiences with you.

2. etwas umtauschen = (to) convert / (to) exchange

Example: Ich tausche meine Münzen in Geldscheine um. I convert my coins into paper money.

Also: Geld umtauschen = (to) exchange money (e.g. at a bank)

and: Waren umtauschen = (to) exchange (unwanted) goods

3. etwas auswechseln = (to) replace sth.

Example: Ich wechsle die kaputte Glühbirne aus. I replace the broken light bulb.

There are some which are good to know:

a) frisch = fresh (e.g. fresh fruit = frisches Obst)

b) ein frisch verheiratetes Paar = a couple newly married

c) ein frisch verliebtes Paar = a couple newly in love

d) sich frisch machen = (to) freshen up

e) jmd. auf frischer Tat ertappen = (to) catch so. red-handed

Those two verbs are tricky, because they’re very similar and are basically synonyms. It’s also super difficult to explain the slight difference, so please be lenient ;):

1. wechseln

Something is replaced with something else. Also abstract nouns or things that don’t belong to you. Usually only one person is involved.

Example: Ich wechsle Geld. I exchange money.

Ich wechsle den Sitzplatz. I change my seat.

2. tauschen

At least two persons have to be involved. They trade one thing for another.

Die Freundinnen tauschen ihre Kleidung. The friends switch their clothes.

Peter und Paul tauschen ihre Sitzplätze. Peter and Paul swap their seats.