For me it's not so much the shoes themselves that make a difference - it's a good pair of insoles. Your feet are almost certainly shaped differently than mine, so rather than recommending the brand and type that work best for me, I recommend going to a specialty shoe store (not just the shoe department at your local Target, but a store that actually specializes in shoes and insoles) and work with them to find a good set. It really helps a lot.
Lots of good suggestions on this thread.
It would be wise to see a physician and get checked out. Too many treatable causes for foot pain, and some of them should be monitored. Is it arthritis? Tendinopathy? Nerve impingement or peripheral neuropathy? There is a long list of potential causes, and the recommendations will vary according to the cause.
I do agree with going to a specialty shoe store, and you might also consider going to a specialty running store, as they will often have employees who are trained at looking at feet and gait. Off-the-shelf insoles work for about 90% of people, although about 10% will need custom insoles.
Nothing wrong with trying a soft mat.
Compression stockings are worth considering, although if the problem is arterial insufficiency (decreased blood supply) compression could make things worse, so go get checked out.
Strengthening the muscles of the foot is a legitimate way to treat quite a few foot injuries, and has recently been recognized as an important approach to rehabilitation of foot injuries. “Towel toe curling” exercises are one way to accomplish this.
Full disclosure: I’m a semi-retired sports medicine physician, so I’ve seen a lot of feet! Good luck with yours!