Answer for My Airport Express (first gen) router

Yes, I would suggest upgrading your router. As far as I know, the newest Airport routers should work with even the oldest iPhones and iPod touches. Regarding wireless technologies, those devices really aren’t ancient!
Let me know when you get a new router and if it helps out. I know the newest Airport routers can easily handle 100 Mbps + connections so I would expect you should see much better performance.