Drugstores such as Walgreens have an expensive large format printer but only stock glossy photo paper. Wal-Mart does photo printing too.
Some art looks good on glossy paper, but if you want matte go to the "office" chains.
Personally I own an inkjet printer and make decorative cards (both photo and art reproduction) regularly. I started the hobby with basic printers and came to the conclusion that: (1) a cheap inkjet printer does great work with quality paper and ink (don't mess around with off-brand ink), (2) if you print yourself you get an element of control that you don't get from the store. (Particularly if you want to get borderless prints just right)
I got my first printer for free until the pick roller gave out, bought an ET-4760, discovered that the ink fades rapidly in the sun, then I upgraded to an ET-8550 because of the better ink and ability to feed through the back which means no fighting the pick roller when you use particularly thick or thin papers.
Consistent quality printing comes from working with a printer…from a relationship.
Otherwise you get what you sent because ordinarily printing is a commodity service, and a business survives on commercial work not the occasional amateur.
Which is to say that getting what you want requires knowing what you are doing, paying to learn by making lots of wrong prints, or paying someone enough to care (and enough to educate you).
The first option might be off the table based on your asking the question.
The second and third options are most likely available with your local print shops.
Knowing the first name of your of your printer is perhaps a minimum for getting consistent art prints. Good luck.
I've also used White Wall [1] in Germany with similar results. They use basically the same process, but an unlicensed version. Bumblejax [2] in Seattle is also great.
The other thing you can look for is local camera stores that still develop film. Chances are they'll have a high quality printer and paper. Giclée printing (basically very high quality inkjet) is what's most popular right now and supposedly is the most durable, but I personally prefer chromogenic (C-type) prints which is where they use lasers to expose photo-sensitive paper. There aren't many places doing C-type prints anymore, so that's generally a mark of quality.
[0] - https://reedphoto.com
I had some prints done in the UK, onto archival paper, the print quality was fantastic. Depending on the source image your milage may vary though. I had prepped my own imagery and soft-proofed them prior to printing - I'm a designer so have a good understanding of what I'm doing.
I'm planning to buy my own large format printer used on craigslist and learn to print large format myself. These printers are really expensive, and the heads can get gummed up easily but I'm hoping there's a way to make it pay for itself (renting it to friends, etc).
For anything smaller, you are probably better off finding a local photography printer and paying them, or just go buy an inkjet.
It wasn't cheap but the results were worth it!
I think asking on the dpreview.com forum for printing, or the photo.net forum is likely to get you good infirmation, from people who care about printing photos rather than signs.