Tesco or Asda??

Asda don't have a loyalty card but are considerably cheaper so you will probably save more money and won't be restricted as to what you can spend your money on unlike the clubcard.
Personally I find Asda food to be better quality although don't buy asda smartprice cheese. It depends what you like it's probably worth a shot to do a shopping list for both stores and go on mysupermarket.com and work out how much you'd save if you went to asda rather than tesco (or if you save anything, depends what you buy).
Asda do seem to have better deals than Tesco and Tesco has been noticed by many in its price rises.

It's really up to you, worth trying different shops though. It's not going to harm you

If you shop in Tesco cos it is near your house then you are missing out. Unfortunately so is my butcher and my baker, because Tesco is the one place I avoid and they are close to my home. My supermarkets of choice are Aldi and Lidl now, and my bank balance is so much healthier than it was before. I wish people would stop crawling the isles of Tesco and Asda for silly things like 6p Andrex and try to work out how brands have cost so many of us so much money.

I too have to consider petrol, but I still think it is worth the effort to do a non brand shop and compare costs. I am still working on the snobbishness of my children who are a bit iffy about it (although I do not include you in that in any way). If you can do the wine, take your time and see what you save.

I regularly shop online between Asda, Tesco and Sainsburys. I have to say that Asda are not always on time with delivery as on a couple of occasions they have arrived after the allocated 2 hour time slot, they also left a bag with dog food (not mine). Tesco always on time and after you select a 2 hour time slot say 2-4pm they will text you in the morning to tell you within the hour say 2.15-3.15. With Sainsburys you select a 1 hour time slot. Sorry, going off a bit there, but as to saving money I do find Asda slightly cheaper and of course depends on what you buy. The delivery costs with all of them are cheaper early on in the week so look at their websites as you could save money there. Hope that helps.

Tesco for range of products, and quality of their Value range, Asda for actually having the products in stock (Tesco seems terrible for running out of stock).

Both seem the same pricewise, and both seem to have lots of stupid pricing tricks, as bad as each other generally.

Personally, I shop there only for the odd item if I'm nearby, preferring Lidl (handy cos its just down the road from my apartment, have a small Co-op even closer, but 57x the price for the same tat lol), Cooltrader, Home Bargains, Farmfoods (fantastic little shop, dirt cheap for branded and quality stuff), Iceland etc...