OK. Now you have been specific enough.
Let's start from very beginning. This forum is about BTC so I'll focus on this coin (but most altcoins are more or less similar)
About wallets:
- best (but not most convenient) is always use the original client for each coin. Problem is you have to download entire blockchain ( which e.g. in case of BTC is 180 GB at the moment).
- most secure are so called hardware wallets (search for ledger wallet that one supports quite lot of altcoins)
- more convenient are so called spv clients ( for BTC e.g. electrum, mycelium, ....) ... these i suggest you to start with
- most convenient are multicoin walltets (e.g. coinomi, exodus ...) which supports vast amount of coins but at some security costs.
check this wallet overview
https://bitcoin.org/en/choose-your-wallet About purchasing:
This is a bit tricky. You have to use exchange and there are two basic types.
- exchages that supports fiat (e.g. coinbase) ... with these you mostly have to verify your identity and are not supporting much coins.
- exchages that does not support fiat (e.g. binance) ... these supports lot of coins but you cant directly deposit fiat
some major fiat exchanges are bitstamp, kraken, coinbase, cex ...
major non fiat exchanges are bitfinex, bittrex, poloniex, cryptopia ...
(check each for it's fees and minimal deposit/withdrawal amounts)
I suggest you to visit
https://coinmarketcap.com/ (search your coin and check market tab for supported exchanges)
To your questions:
Yes, exchanges will hold your coins (no need to have own wallet) ... but be aware exchange could get hacked in which case you most probably lose the coins ... so it's up to you.
Yes, each coin has it's own fee for transfering between wallets. e.g. BTC fee is currently very high and definitely not worth transferring smaller amounts ... better to use e.g. LTC ... it's much quicker and cheaper)
To sum up: I recommend you to purchase LTC (via any fiat exchange) move them to non-fiat exchange and purchase your desired coins.
Alternatively you can visit this section
https://bitcointalk.org/index.php?board=53.0 and try to buy coins from other members.
Hope i did not overwhelmed you again.