All of the wallets I mentioned are iOS apps! That's how I do it!

Again, to buy BTC with funds from your bank, I would recommend a wallet like Lykke, Abra, or Gemini.
You can deposit USD or the fiat ("normal" currency) of your choice and then convert in-app to BTC.

If you want to start trading between cryptos, you can do one of two things --

1. Deposit your newly acquired BTC in an exchange, like Poloniex, Bittrex, Kraken, Liqui, etc. (they are all about the same, but the currencies they allow you to trade vary) and place a trade as you would a stock, etc., with the typical online discount broker.
2. Use the in-app conversion feature, if your wallet has it. Abra does not work with other cryptos right now. Lykke does. Jaxx, which I mentioned in the other reply, does, too, although you have to convert fiat to BTC elsewhere and then deposit the BTC into your Jaxx wallet. Within Jaxx you can convert between a bunch of different currencies, as I laid out, using Shapeshifter, which is pretty much an exchange, built in.

If you go the route of 1), you should be aware that it is recommended to remove your monies from an exchange, depositing them into a wallet, as exchanges are far less secure than wallets.

I trade on Bittrex and Liqui if I want a coin that is not available for trading w/i Lykke or Jaxx. If that coin cannot be stored in one of those two (such as ANS Antshares), then I have to keep it on the exchange or look for another wallet that works with that coin.

For now it is a complex process. AND THAT IS OUR BIGGEST ADVANTAGE. If you think BTC is pricey now, when the proverbial grandmother can do it, the prices will be stratospheric.

Good luck!