I would certainly agree on the casting lessons, look for a GAIA or AAPGAI fly casting instructor local to yourself. Casting lessons aren't particularly expensive and are the best investment you can make in fly fishing as a beginner. Have a chat with the instructor regarding rods and flies, I would expect he'll fish the local ponds and rivers so will be well placed to advise.
For fly lines look at Barrio fly lines as they're excellent value for money. Oh, you need a Roman Moser braided loop too. Personally I'd go with a standard weight forward line for a lake, which would be perfectly suitable for a river but I prefer his small stream lines for river fishing. I would stick with a floating line to start with as they're easier to cast, if you need a little more depth then a poly-leader can help.
IMO reels are massively over engineered for fly fishing, they're mostly expensive line holders, for freshwater fishing a drag simply isn't necessary so just buy whatever reel takes your fancy that's in budget. Reels come in line sizes so just buy the correct reel size for the rod, some people believe that reel weight is important to balance the rod but personally I prefer the lightest reel possible.
You could go for a 9ft 5wt rod, it's a comprise as it's a hint heavy for the river and perhaps a hint light for lakes but it'll work just fine, my first rod was a 5wt. If you can afford two rods then I'd look for a 8'6" 3 or 4wt for river work and something like a 9'6" 6wt for lake stuff. Fast actioned rods require good timing, a good casting stroke and they're also difficult to feel load for beginners too so look for a medium fast actioned rod.
Is hard to advise on flies without knowing the river you're fishing but for me the following are indispensable and probably catch close on 90% of all my fly caught fish.
Tan Klinkhamer # 14,16 - be careful they need to be on the right hook and the right hook is large for the size,
Grey duster or better still the dirty duster #16,18,
CDC & Elk hair caddis #12,14,16
Greenwell's glory #14,16
F-fly #14,16,18,
IOBO #14,16,18,
Tungsten bead headed gold ripped hares ear - 16,18
Tungsten bead headed pheasant tailed nymph - 16,18
Tungsten bead headed red tag - 14, 16, 18