Correct. Read this thread as well:
[General] Bitcoin Wallets - Which, what, why?.
Those are probably signature spammers or don't know what they're talking about. Don't listen to them.
I would prefer an offline wallet, is there one that I can store different kinds of currencies?
I don't think there is one (AFAIK).
How hard is it to make online purchases at exchanges using an offline wallet?
What are 'online purchases' with a 'offline wallet'?
I don't see any reason for me to have to download GBs of data to have an entire blockchain, I'm assuming it is not necessary? I was reading about an ethereum wallet that seemed to require it.
It is recommended to use a full wallet (~100GB of blockchain right now). However, this is not necessary. You can use an SPV wallet such as Electrum.
Finally, which exchange should I buy the currency from?
Depends on which region you're located in and what exactly you want to buy.