If that is the case then use web wallets instead. IMO they are the best ones to use for payment processing everyday. I recommend you use blockchain wallet, it is one of the most convenient web wallet out there that I trust. There are more mobile application btc wallets but like what the post above me said, I also recommend Copay. IMO they are one of the best mobile wallet made for android.
I disagree. Lightweight desktop wallets like Electrum are just as easy to use, charge much lower fees (you can choose your own mining fee), and are much more secure as you actually have sole control over your private keys. If you're a newbie, I would strongly recommend it.
I'm concerned on a number of issues the main one is loosing money. I'm not really investing money in Bitcoin but using it as a means of payment. $500 worth. Does a different wallet really download quickly? I am five days into downloading Bitcoin Core Wallet and it isn't 50% done and it says 2 more weeks which will probably get longer the way this has been going. All this is very new to me and I'm taking a leap of faith to get this working. I still haven't made it to steps where I buy Bitcoin or spend bitcoin. This is kind of nerve racking. I definitely might not be cut out for this. Trying to conceptualize all this from start to finish when I haven't done this or seen it done, well I'm sure you know.
Core is that way because you're also downloading a copy of a blockchain, and it sets up a node itself. Few people actually do this (but more should). As mjglqw has said, you should try other desktop wallets like Electrum, which is a lightweight software and should be up and running in just a couple of minutes. There's really no technical barrier for entry for Bitcoin, so don't get discouraged!