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