Disclaimers: I am new to this. Most of my store trips to date have been for small quantity stuff, not for re-doing an entire house. I am not affiliated with any companies mentioned.
In Kansas City, HD and Lowe's are the big players. HD was here first, and Lowe's showed up soon after. There is a Menards in St. Joseph but that's 55 miles away; I think there is a much closer one in the works, but it's not here yet.
I've shopped at Menards in Iowa a couple of times, but not enough to reach a general conclusion. Their selection *looks* more extensive than HD or Lowe's, but I also think some of that is down to how Menards builds their in-store displays and shelves.
Usually I find HD has better prices on the same material. I also find that sometimes HD has the more well-known/older brands, while Lowe's has the "me too" brand of a particular product.
I think HD is better for "stuff you can't see" - lumber, screws, nails, rough plumbing, rough electrical, and so on. Lowe's is sometimes better at "stuff you can see" - light fixtures, faucets, flooring, curtain rods, and so on. Lowe's has more choices, especially at the higher end, and more up-to-date styles.
Sometimes HD's lower prices work against them - if they are sold out, I can go to Lowe's and find the item on the shelf, for those times when having it today is worth a 5%-10% price premium.
Around here, HD also has tool rental (in some stores), and truck rental. Lowe's only has truck rental. If it's a job that needs a rental tool, it's easier to get the tool and all the other pieces at HD.
I don't usually shop at hardware stores like Ace or True Value, except for certain things. The hardware stores are a lot better if you just need a few of some goofy fastener; they have more selection and will sell you 1 at a time. They are also a little better for when you want to buy just the guts of something like a faucet and rebuild it, rather than swapping the whole thing. Related to that, the repair parts that HD and Lowe's do stock tend to be only for relatively "recent" stuff, like what has been common for the last 10 or 20 years... if you have a 50 or 80 year old house, you're more likely to find parts that fit at Ace or True Value, compared to HD or Lowe's.
Some of the hardware stores make it a point to hire retired or part-time tradespeople, which can be handy if you need some advice to go with the goods. I haven't seen this as much at HD or Lowe's, but sometimes it happens.
Every once in a great while I buy something at Wal-Mart. Their selection is not extensive, but they are open 24 hours a day, and the hardware aisle (in a Supercenter) has a lot of stuff that might need to be fixed right now - sink drain traps, toilet parts, switches, outlets, cover plates, door locks, etc. The prices are at least close to HD, but if I'm buying it at Wal-Mart, I am probably not worried so much about the price.