I own, and love, a fancy printer (Canon Pro 200). As well as a number of non-fancy printers (Epson XP8500, Canon SELPHY, INSTAX).
But if you’re happy with the print quality you get from the shop, and the paper choices they give you, and don’t need prints right away; then a print shop is simpler and cheaper.
Printers don’t like to sit idle for long times. After sitting idle for too long, they will need at least a cleaning cycles (which wastes ink), or they clog and need to be unstuck. At least that’s what “they” say. My inkjets never needed more than a cleaning cycle, even after months of disuse. The higher end pigment printers may be more capricious.
For up to A4, a simple six-ink desktop printer such as an Epson XP8700 is not that expensive, and prints a similar quality to a normal lab print. Inks don’t last forever, but they’re relatively affordable (€100 for a full set), and the printer itself is affordable, too. Ink and paper probably won’t be much cheaper than the lab, but you get to print at home and experiment with stuff. I’ve used my XP8500 for many years, and the photos come out great. They last many years, too. Perhaps not “archival” quality, but easily good enough for a few decades.
For A3 prints, my Pro200 produces better prints than a lab, at full archival quality. But neither the printer nor the inks are cheap in any way. So that’s probably not the right route for you. It’s one of the least expensive top-quality printers, though, if you don’t print large volumes and don’t require pigment inks.
The other option is a photo-capable inktank printer, the Epson ET-8550. Here the ink will last so long, so as to no longer play a large role in your price per picture. It probably produces fractionally nicer black and white prints than my Pro200 (more black/grey inks), and maybe slightly lesser color prints (fewer color inks). But probably easily on the same level as a decent lab. But the printer itself is expensive.
Keep in mind, good photo paper is expensive, too. You’d be hard pressed to save significant money over the lab prints just on paper cost alone. Factor in inks and the printer, and home printing just doesn’t make much sense financially.
But you do get to play with various paper stocks, you get to experiment with layouts and things, and you get to do it at home, without waiting for shipping. For me, it’s a thing I enjoy doing, and that makes it worth the cost.