According to the Walmart web site, your bike already has 7-speed Sora brifters. However, ALL Sora brifters of any "speed" have the thumb shift button so changing for newer Sora gets you nowhere.
You would have to go to Tiagra or above to avoid the thumb button and that would also require a new cassette, new rear hub and new chain since current Tiagra is 9-speed and Shimano's higher groups are 10-speed. All 7-speed groups are obsolete. It will be big $$$.
I agree that upgrading this bike would be throwing good money after bad. Ever heard the cliché about making "a silk purse out of a sow's ear"? Ride this one to be sure you like road riding and put up with the current shifters. When you are sure you do, save your money and buy a worthwhile bike from a real bike shop.