Best Crypto Wallet for Alt Coins

Hey guys,

I recently bought a few alt coins on exchanges such as cryptopia and Kucoin and Binance. Basically I'm trying to find the best multi wallet where I can store all these coins or do some alt coins have to have separate wallets.

Also - if the value of the coins increases (so Im making money) on the exchange and I transfer to the wallet will the value still go up? if that makes sense. Or would I have to leave the alt coins in the exchange to actually see increases.

Thanks a lot!

Adi