PCT Thru hiker here, so I have a wee bit of experience on hiking.....
OP - As others have noted, I'd HIGHLY recommend that you start off by renting gear. IF you then decide to keep on backpacking, then start buying gear.
2nd hand is the lowest cost way to go. Here in Seattle, there is ALWAYS going to be quality 2nd hand gear for sale. Even if you buy new, there are places to go low cost / low tech, and places where for long term value, you want to get the highest quality you can afford.
The Big 3 - pack, sleeping gear and shelter.
The pack is the last thing to buy, as it's merely the container with which you carry the other, useful gear. Size it to carry what you have in the way of other gear. A cheap, used, external frame pack can likely be found on Craigslist. 60 liter is a fair size if you're reasonable with your other gear.
Shelter - it depends on your circumstances - that is, are you planning on backpacking in an area with minimal chance of rain, with no bugs? Hey great, a tarp is all you need - make your own from some plastic sheeting and parachute cord. If you're in bug hell Oregon in July / August, a fully screened shelter is probably the better choice for your sanity. The rainy PNW in the shoulder season demands a bathtub bottom shelter, with good sized vestibules and bombproof rain protection. For a used mainstream tent suitable for backpacking, see if you can find one of the 2 person REI dome models. They're solid, good rain and bug protection - not the lightest, but light and compact enough for backpacking, but generally lower cost than a lighter / more compact, higher end tent.
Sleeping gear - since you're backpacking, weight and volume are a consideration. Here's where, once you've decided you're going to buy, I'd recommend getting the best quality you can afford. If you can score a used down back with 800 or higher fill from a reputable maker, get it - Marmot, Western Mountaineering, Feathered Friends, the mid / higher end REI house brand, or equivalent. If you show up and the seller pulls it out of a stuff sack, walk - the down is likely ruined from being stored compressed. If it was stored hung up, or loose in the large cotton bag, that'll be A-OK. for the sleeping pad, the least expensive are the foam "Z" folding, or a Ridgerest, or a Green flat Army Surplus style. Remember, a pad is for insulation from the ground, comfort is a 2nd and lesser consideration. Set up on soft ground (thick layer of duff) for comfort.
Other gear - Cooking. If you don't live in a fire prone area, like So Cal, then a home made alcohol stove is the cheapest, lightest way to go. Google "Pepsi Can Stove". Or cook on Esbit tabs. If you are in a fire prone area where you must have a shut off valve - something like / similar to a Pocket Rocket is a great choice.
Cook pot - Lots of folks recommend a "Walmart Grease Pot" for a cheap, aluminum container suitable for boiling water in. Looks to be about $6. If you use Esbit tabs, a Fosters "Oil Can" with the top cut off / open is a suitable pot to boil water in. Spoon from home = free.
Headlight - Low end Petzl from REI or similar.
Clothing - Synthetic base layer, convertable nylon pants. Cheap fleece layers bought from Goodwill / thrift store. In a crunchy Granola town like Seattle, lots are available.
Rain gear - A poncho is the lightest, best and cheapest upper body rain gear. Get one of the nicer ones (not the emergency kind), which for $20-$40 will last you a lifetime, and also cover your pack, eliminating the need for a pack cover. A trash bag that you slit the bottom out of becomes a rain skirt for the legs - use the drawstring type to facilitate tying around your waist.
Footwear - keep your pack weight reasonable and just use your running shoes. Boots are so 1990. Nearly all thru hikers use running shoes.
Water bottles - forget nalgene's. Expensive and heavy. Buy a $1.59 1 liter bottle that comes free with a liter of soda at the local quickie mart. Drink soda, rinse, tear off label - you have one bottle. Repeat 2x times (note that 1 liter gatorade bottles also work well). Supplement with a Platypus bladder if you're desert hiking and need lots of water capacity.
Water treatment - Aqua Mira drops gets 'er done cheap in 20 minutes. I went through about 4 kits in 2600 miles of PCT thru hiking.
Food - MRE's suck - they're heavy and expensive and generate a lot of trash. Dinner: Knorr brand rice & pasta sides at the local grocery for dinner. Add Instant Mashed potatoes to fluff up the volume. A Deluxe Mac and Cheese with the pre made sauce is a good choice as well. Foil pack / canned chicken, tuna, salmon for meat. Lunch - bagel sandwiches with dry Italian Salami and cheese harder than a medium cheddar (note those don't require refrigeration even in 100 degree heat for 4 days - I know from experience, they have enough salt and not enough moisture - will get a bit greasy, but won't poison you). Breakfast - oatmeal with powdered milk, Via for the coffee. Snacks - lots of nuts, granola bars, Hostess Fruit Pies, Ding Dong's, snack cakes, etc. Thru hikers get ALL their food from local grocery stores (except those on special diets where they make and have stuff sent from home).