If your Target has a tiny grocery section then your Target must be a regular Target instead of Super Target.
I find Wal-Mart to be a beat down so I don't shop there. It's too big to easily take the boys without worrying about losing the toddler or one of them having a melt down.
For quick trips for staples with the exception of produce I go to Target since it's close and I have a debit card that gives me 5% off and takes the money directly from the linked checking account. The produce selection and quality stinks though although I like their store brand.
For fruit, diapers and wipes, organic milk (one of the only things I always buy organic), household cleaning supplies, over the counter meds, shampoo, almond butter (both boys' schools are peanut free and almond butter is costly anywhere else) and a few other things I go to Costco. The prices and quality are great plus I can return anything including opened items without a receipt. The only problem is that there are so many temptations there I really have to police myself to stick to my list.
For regular, long grocery lists I go to Kroger. There's one 10 minutes away that has a really great produce department plus they sell amazing store made tortillas. They also make their own sushi in house (you can watch them make it) that I rarely buy for my older son as a reward for good behavior. They also have car shaped carts which the boys will happily share. That makes it easier to take them both shopping at the same time.
I don't have time to visit a whole bunch of places to get everything I need so Aldi's is out. I got tired to going to Aldi's and then having to go to the Tom Thumb across the street to pick up what Aldi's doesn't have. I'd have to say the same thing about Trader Joe's and Sprouts.