Loads of choice for travel fly rods, many of the top end rods now some in 4pce as standard...........how they have come!
Within your budget I have good experience with Greys and fulling mill but worth looking out for are Cabelas own branded ones on Ebay.
I would try and get the best rod you can with a fly-rod, it makes the whole experience more enjoyable, less tiring and the rod would last longer. My choice would be Greys because of the warranty although I would guess you might need to spend a little more, or just go for it and get a missionary or an Orvis Frequent Flyer!!
Reels is kind of confusing as traditionally (and still today by some) anglers would cast with their right hand and also wind in with their right, crossing hands on the basis that they wanted their fastest handto recover the line. I never got that. It has also been said that the reel is just for storing line, I don't go for that either!
I cast with the right and wind with the left (this is L/H wind), I also use the reel when playing fish as its so easy and reliable with modern large arbour reels and they have the ability to be free running on the line recovery so any spare line can be panned back and you're direct contact in a breath.
If you need advise on a reel, I use the Greys GRX, superb and come with plenty of spools and extra ones are cheap as!