Assuming you are talking freshwater, 5 wt rod is fine. Don't spend too much - you can get something from Temple Fork, Readington, St. Croix or Albright might be good places to start. You can get a decent rod for under $200. Don't be talked into this line of reasoning - spend the money now and get a great rod that you will appreciate for life...you will be a couple years away from being a good enough caster to appreciate a top end rod, and with a new baby (and an empty flybox...) you have other things to spend money on. Also - try the rod in the parking lot. And not by seeing how far you can cast. In the NE 90% of your freshwater fishing will be within 50 feet max. So being able to cast the whole line is jsut a silly exercise. Better idea - pick a couple spots on the pavement at different distances - 20 feet, 30 feet, 40 feet - now alternate casting to these spots. You may find that a "cannon" is not so good at this. Try fast, moderate and even slow rods - you will probably end up with a moderate rod. Go with a minimum of 8.5 foot long - 9 ft is probably optimal. Some discount rods do not come with tubes - get one.
Reels - lots of good reels around - you don't have to go crazy. Orvis Clearwater is good starter. But another thought is to see if where you are buying the rod has a starter set with rod, reel and line.
Line - 5wt, weight forward floating. And get a brigth color - it will be easier to learn to cast with.
Waders - don't get hippers. Get waders. The breathables are the way to go. Orvis, Patagonia, Simms - all have low priced options. And get stocking foot ones - you will need boots too, but they are much more flexible, more stable when wading, etc. Walk around in the store in them, and try stepping up onto chairs (like climbing out of the river).
Tippets - 4X, 5X, 6X - get a spool of each. Keep out of your trunk or sun.
Flies - buy 'em when you need them, from a local shop based on what is needed locally at the time. Then you will not have a selection of flies that sit in your fly box and are never used. Keep them in one box initially - the cloth covered foam ones are good starters. Others are better, but you can add them in later.
Miscl - selection of tin split shot, indicator putty or yarn indicators, clippers (get stainless ones), and from a local hardware store get forceps to get hook out of fish's mouth for about $5. Have decent polarized sunglasses - helps to see fish, as well as the log you were going to trip over. Vest - try it on with sweaters on, and pockets filled with boxes (just borrow some at store) - maybe to start, just use a fanny pack moved around to the front, and skip the vest. Net - you can get a decent one for maybe $50 or so - wood frame, and catch-and-release netting. Discuss at shop how to hang on back of vest. A thing of Gink or similar fly floatant.
Lessons - first two options to look at - check out your local Trout Unlimited chapter - best $15 you can spend to join (and includes a magazine!). They often have periodic classes in casting, as well as fly tying for those inclined. Monthly meetings generally. Just search on Trout Unlimited on the web - you can be directed to a local chapter. Also, shop where you bought rod may have classes. Idea to get feel of casting - don't start by casting back and forth over your shoulder - run 15 feet of line out from the rod tip, hold the rod in front of you, and move it horizontally back and forth in front of you - cast to the left, and see where the rod goes. Now right. What happens if you stop and start your cast crisply vs. accelerating and decelerating more slowly? Now take your wrist out of it - cast with your arm. Get used to this wrist/arm thing. Now try to cast over your shoulder and feel the same stuff.
Fly Schools - I went to the Wullf School a long time ago (Lee was still alive), and they are great experiences. But maybe to start, play around a bit and you may get more benefit.
For now:
1. You have a new baby, and likely a stressed Mom. Don't disappear for 12 hours at a time in the Catskills, or she is going to resent fishing. For now find some local waters - NJ has lots of them. A local pond with sunfish is frankly a lot of fun, and a great way to learn casting, setting the hook, etc. without worrying about the current carrying around your line, or wading. Lots of trout streams too - see your local fly shop.
2. Plan on taking the kid fishing beginning at around age 3 - worms and sunfish are best, esp. in spring when they are bedding - kids like action, and love catching fish - who cares about the size. Tell your wife of your plans now, and when you find a good pond to take the kid when they are older, tell her now. You are not fishing. You are looking for places to spend quality time with the kid to give your wife a break...Started my daughter at 3 (actually had a rod for her under the Christmas tree before she was born...), got her waders at age 7, and her first trout on a fly she tied herself being an 18" brown. Here's the deal - "Honey, have to go up to the Beaverkill for the evening with Caitlin. See ya!" Warning - you may find yourself teaching flycasting to a bunch of Cub Scouts...