Insoles matter more than the shoe itself, honestly. You can grab a decent sneaker and drop a pair of good arch support insoles in there - something like Superfeet or Powerstep - and get way more relief than waiting for a specialist fitting or hunting for the "perfect" shoe. The insoles do the actual work of supporting your arch, and they're way cheaper than replacing shoes every few months.
That said, if you're starting fresh with sneakers, look for brands that don't fight you. Nike, New Balance, and ASICS tend to have options with stiffer midsoles and built-in structure that play nice with arch support. Casual sneakers from most brands usually have soft, flexible soles that collapse under your foot, which makes flat feet worse. You want something that keeps its shape.
The real move is to get insoles first, try them in whatever sneakers you already own, and see if that kills the pain. If it helps but isn't enough, then you know it's worth upgrading to shoes with better structural support. Trying on shoes at a specialty store is fine if you want confirmation, but it's not required - most people figure out what works by trial and error anyway, and insoles are cheap enough to experiment with.