Hiya, neighbor, welcome to Logan Square!

I live right by you, so I hope I can provide some insight. For the most part, these are all very, very budget friendly options in the immediate area. I have a car and am on a budget, so I personally divide my shopping dollars between Tony's, Costco and Aldi, with side runs to Strack and Armitage Produce as needed. While those are my personal favorites, once you have had a chance to explore a bit, I'm sure you will have some new gems to add to this thread!
Although there is no dedicated thread I could find (note - I fail at the search function on LTH)
Tony's Finer Foods is mentioned everywhere and is a really excellent Mexican oriented grocery store you can walk to. Although it is primarily a "Mexican" grocery store, they cater to everyone in the neighborhood and carry a large variety of standard and other ethnic (Polish/Asian/Italian) items and at very reasonable prices. Great and inexpensive produce, a vast and affordable meat section - with actual butchers - even a decent booze aisle, although with VAS so nearby, you may just do your alcohol shopping there. Bring a quarter for a cart (set up like Aldi's) or you will wind up lugging an impossibly heavy basket throughout the store. BTW, it is insanely cheap and everything I've every purchased here was high-quality.
Tony's Finer Foods
3607 W. Fullerton Ave.
Chicago, IL. 60647
773-278-8355
A very nice Puerto-Rican oriented grocery store that is walkable is Armitage Produce . Although I find the selection, all around, to pale in comparison to Tony's, it's still a fine market with virtually every Goya product every made and a very interesting produce and meat department. As it skews PR, expect to find more items like Bacalao and fresh Gandules. It is a very bright and clean market and when I'm feeling too lazy to walk or drive over to Tony's, I gladly walk over here although I find that it is a smidge more expensive than Tony's.
Armitage Produce
3334 W. Armitage
Chicago, IL 60647
773-486-8133
There is also a nice Panaderia right by Armitage Produce called La Estrella. Language can be a barrier here if you have questions, but I typically just grab my tray and tongs and am set to go. Very nice and inexpensive loaves of french bread here daily along with the usual colorful selection of baked treats. As it is located on the way to Armitge Produce, it makes for a nice circuit.
3309 W Armitage Ave
Chicago, IL 60647-3717
(773) 772-1861
Further afield, but still within walking distance, are Panaderia La Central and a little Mexican market (next door to the Cozy Corner across the street from La Central), Emilio's Produce. Both are great places to stop to pick up the ingredients for dinner if you happen to get off at the California L stop. I've had very good luck with both the meat and the produce at Emilios, particularly the meat with their very friendly butchers.
Vas Foremost is also over here and is really a more than you would expect liquor store. They are quite accommodating and happy to special order. This is also fairly close to
Provenance, a nice little cheese/wine/gourmet shop a bit further north on California. The owners are just lovely and they always have a good selection of $10.00 or under wines. If you find yourself needing, say, fish stock or Mozzarella di Bufala at the last minute, this is probably your best bet.
Panaderia La Central
2218 N California Ave
Chicago, IL 60647
(773) 235-8677
Emilio's Produce
2294 N Milwaukee Ave
Chicago, IL 60647
(773) 486-5905
Vas Foremost
2300 N Milwaukee Ave
Chicago, IL 60647
(773) 278-9420
Provenance
2528 N. California
Chicago, IL 60647
773.384.0699
A bit further still and perhaps more bus destinations in or near the neighborhood are
Strack and Van Til, a large and diverse supermarket with very good quality and attractive prices,
Carniceria Jimenez, another really fabulous Mexican grocery store,
Aldi, a person on a budgets best friend, and, of course,
Costco if you happen to have the space to accommodate buying in bulk.
Strack and Van Til
2627 N Elston Ave
Chicago, IL 60647-2018
(773) 252-6400
Carniceria Jiminez
3850 W Fullerton,
Chicago, IL 60647
773-278-6769
Aldi
1767 N. Milwaukee
Chicago, IL 60647
Costco
2746 N Clybourn Ave
Chicago, IL 60614-1006
(773) 360-2053
The bodegas in this area swing wildly from uber-sketch to nice family run establishments. As in any Chicago neighborhood, you'll get to know the ones nearest you. Overall, for convenience, I can recommend Food Smart. It's a much better than average mini-market and they have recently greatly improved the quality and breadth of their offerings to include fresh, organic meat, high end cheeses and craft beers. They have a bit of everything and it's really an easy stop when you don't have the time/energy to get to a larger store. They also stay open quite late. Of course, much like the neighborhood bodegas, everything is priced a bit higher here, so you do pay for the convenience.
Food Smart
2901 W. Armitage Ave.
Chicago, IL 60647
(773) 772-3200
I'm sure others will chime in with their favorites, this are just the places where I shop regularly directly in, or, very close to the neighborhood.
"Baseball is like church. Many attend. Few understand." Leo Durocher