Translation:We buy new rackets for the children.
I think I remember a similar sentence about buying something for someone and the preposition "åt" was used. Does it mean the same?
Yes, both till and åt work for the case when you buy something in order to give it to someone, like here.
I think I read somewhere that för would be used if talking about buying something so that someone else doesn't have to buy it (i.e. sort of like doing the buying in their place). Is that true?
åt is the preposition you use when you do something so that someone else doesn't have to.
Jag skriver ett brev till dig 'I'm writing a letter to you'
Jag skriver ett brev åt dig 'Im writing a letter for you' [meaning: so you don't have to write it yourself]
I laugh at you-jag skrattar åt dig.
In that case "åt" is different, it can be thought of as "in (your) direction" (the same way as one sais "sväng åt vänster"-"turn (to the) left").
A few other verbs are similar, like skriker=shout.