Miyawaki has alot more experience than i do with fly fishing but i would recomend you try out a spey rod go cast some at your local fly shop and see what you think. if you do not like the spey i would say go with a TFO 7-9wt. Or you can look on the classifieds here there are always good rods and reels for sale.
A 9 weight single hander would be fine if you want to fish for all three of the species you mentioned. A 7 or 8 would be better for fishing for both reds and silvers. I would go with either a Echo or Redington package for both the rod and reel. You could obtain this under budget and have an awesome starter setup.