I don't recommend buying new equipment for your first few backpacking trips because you won't really know what you want. But if you're ready to make to make a purchase, take a look at what I use.
You can see my full list of gear here.
There are tons of options out there. My choices below are comfortable and lightweight for those who don't want the bare minimum but are also looking for creature comforts. Below are essentials for any backpacking trip: shelter (I like to be enclosed because it keeps my mind at ease about having bugs crawling all over me), sleeping mat (to keep you off the ground = warmth), sleeping bag (I tend to run cold-ish, so I love being surrounded by a fluffy down).
OPTION ONE: Soto Microregulator stove + Snowpeak Hybrid Cookset
OPTION TWO: Jetboil
If you're into taking photos, this is my current backcountry camera system.
There are so many opinions about what to wear. OMG. The main criteria I think about for apparel are: comfort, function, and quality. I don't own a ton of options because I wear the same thing over and over, and mix and match depending on where and when I'm going backpacking. Below are my go-to's.
SMALL ESSENTIALS & ACCESSORIES
These are all the small things that can matter. I'll tell you a bit about each item.
ROW 1 (L to R): Headlamp - Absolutely need! One of the handiest bits of gear you'll need; External battery - I don't need this on trips less than four days, but I'm certain I'll need to charge my iphone, it's one of the best options out there; Mini compass - If I'm not going anywhere complicated, this compass will do. It has a nice clip that attaches securely to my backpack's sternum strap; Pocket knife - I carry one in my pant pocket because sometimes we need to cut things like rope, cheese, and salami - you know.
ROW 2 (L to R): Collapsable water bottle - Love mine! It's so convenient for many reasons; Water bladder - Not necessary. You can also get two 1-liter plastic water bottles; Stuff sack - Not necessary, but I use one as a bear hang; Hiking poles - are AWESOME for taking some pressure off both uphill and downhill; Bear spray - In some parks, it's required.
ROW 3 (L to R): iPod Shuffle - If I'm hiking solo, sometimes I load this little guy up with music and podcasts; Reflective rope - Bear hang must; S-biner - I use this clip for my bear hang or to hang stuff off my pack; Handkerchief - Countless uses (towel, pee-rag, wash cloth, etc)