Here are some solid recommendations from Amazon:

Tent - https://www.amazon.com/Big-Agnes-Copper-mtnGLO-Backpacking/dp/B08175NX4C/

Backpack (mens) - https://www.amazon.com/Osprey-Packs-Atmos-Backpack-Abyss/dp/B077H3G5J4/

Backpack (womens) - https://www.amazon.com/Osprey-Womens-Backpacking-Aurora-Purple/dp/B07GHKJSWD/

Sleeping bag (mens) - https://www.amazon.com/Big-Agnes-DownTek-Sleeping-Regular/dp/B07M9WL54C

Sleeping bag (womens) - https://www.amazon.com/Big-Agnes-DownTek-Sleeping-Regular/dp/B07MH3B1R2/

Sleeping pad - https://www.amazon.com/Therm-Rest-Ultralight-Backpacking-Mattress/dp/B07YFXVVFD

Pillow - https://www.amazon.com/NEMO-Nemo-Fillo-Goodnight-Gray/dp/B083BJH5NX/

Stove - https://www.amazon.com/MSR-PocketRocket-Ultralight-Backpacking-Camping/dp/B07L5S65HR/

Pot - https://www.amazon.com/Snow-Peak-Trek-Aluminum-Cooker/dp/B000AR2N2Q/

Cup - https://www.amazon.com/Snow-Peak-Single-Fresh-Water/dp/B004YOMW40/

Bear can - https://www.amazon.com/BearVault-Resistant-Canister-Backpackers-BV450/dp/B001V9HENU/

Water filter - https://www.amazon.com/Sawyer-Products-SP129-Filtration-Squeezable/dp/B00B1OSU4W

Water bladder - https://www.amazon.com/Cnoc-Outdoors-Container-Thread-Orange/dp/B07QBQ894L/

Headlamp - https://www.amazon.com/Black-Diamond-Sprint225-Headlamp-Ultra/dp/B081B8CXH6

Satellite communicator - https://www.amazon.com/Garmin-Lightweight-Satellite-Communicator-Handheld/dp/B09PSSSFPF

Knife - https://www.amazon.com/Victorinox-Swiss-Classic-Pocket-Knife/dp/B00004YVB2/

Dry bag - https://www.amazon.com/Sea-Summit-Lightweight-Sack-Large-13-Liter/dp/B001Q3KKG4/

First aid kit - https://www.amazon.com/Adventure-Medical-Kits-Ultralight-Watertight/dp/B0009RIQBE/

Chair - https://www.amazon.com/Big-Agnes-HCHAIRONEB-Helinox-Portable/dp/B007ZGOWZQ

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If you don't like those specific items - the brands are all legit, see what other options they offer.

REI.com has a list/registry option too. I recommend it over Amazon - where questionable products are abundant.

Clothes are hard to buy as gifts - but important for backpacking. Patagonia, Arc'teryx, Icebreaker, Darn Tough, and Smartwool make quality stuff.

You could also add a bunch of fancy dehydrated meals like Mountain House, Backpackers Pantry, and Packit Gourmet. Or just flats of beef jerky and dark chocolate.

I'm bringing this. Bought it a few months ago and I love it. Weights an ounce, has knife, scissors, file, tooth pick, and tweezers. The only thing I miss on this one is pliers, but I get by. If you wanna go light this is where it's at

http://www.amazon.com/gp/aw/d/B00004YVB2/ref=mp_s_a_1_1?qid=1459560971&sr=8-1&pi=SY200_QL40&keywords=swiss+army+knife+classic+sd&dpPl=1&dpID=4176SP96teL&ref=plSrch

Big 4 for $502

• Backpack - Osprey Exos 58 - $220 - Link
• Tent - NatureHike Cloud 2 - $120 - Link
• Sleeping Bag - Kelty Cosmic 40F - $130 - Link
• Sleeping Pad - Random Foam Pad - $32 - Link

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The rest of the gear is relatively cheap, like:

• Stove - BRS 3000 - $17 - Link
• Pot - Toaks 750ml - $35 - Link
• Headlamp - Nitecore NU25 - $37 - Link
• Water Filter - Sawyer Squeeze - $30 - Link
• Knife - Swiss Army SD - $19 - Link
• Lighter - Bic - $1 - Link
• Compass - Random Brand - $11 - Link
• Pack Liner - Glad Compactor Bag - $5 - Link
• Battery Pack - Anker 10K - $22 - Link

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Random stuff from your house / DIY:

• Spoon from your kitchen (or gas station)
• First Aid Kit from your bathroom cabinet
• Duct Tape from the garage
• SmartWater bottle from the gas station
• Map printed from CalTopo.com

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Clothes could be a whole post of their own. Start with what you already own and upgrade to wool/synthetics over time.

Food from the grocery store is cheap. Skip the fancy freeze dried stuff. Shout out /r/trailmeals.

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Skipping unnecessary items is the best way to save money. Amazon will make you think that a huge knife, lantern, chair, axesaw, and other 'camping' gear is important. Don't let them trick you into wasting money / weight on garbage.

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EDIT: added a few items