That sounds like a 6.1 arrangement. Though you can always just not use the center back location and do a classic 5.1 setup instead.
Frankly most surround sound is either 5.1 or 7.1, they kind of skipped over 6.1. It won't break anything to do a 6.1 setup. You can of course always do a 5.1 setup and add that 6th speaker later, so long as your A/V can handle a 6.1 setup. Just note that since 6.1 was skipped over that means your options are really 5.1 or 7.1 A/V receivers. 5.1 can never go higher than 5 speakers but 7.1 can do 7 or less speakers.
Just a side note you do not want to buy a system in a box (something that comes with the receiver and all the speakers in one package) I can guarantee you to almost 100% certainty that they will not let you use the prewirred connections your house has. They also are generally the worst quality.
So you'll need to buy your parts separately; receiver,4 speakers, 1 center speaker and a bass if you want. My suggestion is if you don't want to spend a bunch of money right now then get a good received $250-500 and just 2 left/right speakers ($100-200 ea) for a simple 2.0 setup. Will still be worlds better than whatever sound you TV can put out on its own. Then you can just add more speakers over time as you feel like it.
I don't have a recommendation for a receiver. But I do like the speakers Polk audio makes, their prices are decent too.
Oh also be 100% about those speaker wires. Most just use copper wire that you twist around the connection pole. But some brands use these plugs that are custom to their brand only.