Obviously, one of the great appeals of REI is being able to try things on, walk around, talk to the staff, etc, but they do have a website (and an outlet website!) for you to purchase necessities.
Speaking of online, it is your best friend. For all small stuff, amazon will be best (cooking gear etc).
This is a similar tent to the one I use, and I think it's unbeatable for the price: http://www.rei.com/product/864796/rei-quarter-dome-t1-tent-special-buy/?cm_mmc=cse_PLA-_-pla-_-8647960001&mr:trackingCode=52F379B0-6C5B-E411-AF04-BC305BF82162&mr:referralID=NA&mr:device=c&mr:adType=plaonline&mr:ad=55518619600&mr:keyword=&mr:match=&mr:filter=68532232120&msid=yomIsid8_dc|pcrid|55518619600|
Sleeping bag will be tough to find a deal on, but at rei outlet you can sometimes find one for ~80 bucks. I highly recommend a sleeping pad as well. I would definitely go with synthetic over down at your price point.
You don't have to spend a fortune on boots, but if your feet are wet. you will be miserable. I generally buy mine on 6pm.com If you wait for the right deal you can pick up some WATERPROOF Keen/Merrell/Northface type boots for well under a hundred bucks. They won't last you ten years, but they will keep your feet infinitely happier than they would be in sneakers. Last thing on shoes: break them in. Whatever you think is a good amount to break them in, probably double that.
This might be heresy, but I'll say the place you should skimp is on the pack itself for a couple reasons.
1a. If you're going for your first time, you don't even know what you like.
I think trekking poles are for old people.
Here's how I'd break down the $600 if I were just starting out and had nothing.
~150 REI Quarterdome tent ~50 whatever cheapo internal frame backpack strikes your fancy on amazon ~80 whichever sleeping bag you find on sale with the lowest temperature rating. Nights are cold where you're planning to visit. ~40 Whatever sleeping pad you can afford. ~80 Some waterproof shoes and 2 pairs of merino wool socks. The socks are important. Trust me. ~100 Water pump. Iodine tastes like butt, and boiling is a pain. ~100 Jet boil stove. This is the Cadillac of backpacking stoves. Saves space/weight. Saves time. You won't ever need to upgrade. This is where I'd splurge.
Hopefully at the end you'll have a couple bucks left for a spork and some odds and ends.
Last thing. You need some light. Any home improvement store will sell headlamps for about 5 bucks. These will be a reasonable copy of the REI equivalent for 10% of the cost.
Have fun out there.