Budapest is good value for money from a Western point of view, but avoid the tourist traps. One of my favourite restaurants is Koleves on the corner of Dob/Kazinczy utca in District VII, close to Deak Ter. You'll be surrounded mostly by locals. Liszt Ter is nice, particularly in Summer. In September, the weather should still be lovely. It's a bit touristy though and the service is at best leisurely. Also, somebody, somewhere will try to rip you off, so always count your change. Cafe Alibi on Egyetem Ter in District V is great for breakfast.
Art'otel on the Buda side of the Danube for modern style and the Corinthia Grand Hotel on Erzsebet Korut for grandeur are two of the best medium-priced hotels in the city. Make sure not to pay full price though as there are always good offers available on the net. The Four Seasons is beautiful too, but prices are crazy. Definitely worth going to though for a coffee at some stage.
Don't hail taxis on the street. Stay away from anything with a 'tourist menu'. Try to have smaller notes, as restaurants and bars frequently don't have enough change. Buy a transport pass for your entire time here. Go to the Szechenyi Baths in City Park. Walk down the Danube embankment on the Pest side at night. Go the opera house, even just to see it. Don't take the metro everywhere - it's a beautiful city and wandering around is half the enjoyment! Go to some of the outdoor bars if you have time - particularly Hold Udvar on Margit island, which is one of the few that's suitable for families. Szimpla Kert on Kazinczy utca or Tuztate on Tuzolto utca, if you want to see where the younger locals go.