You need a kayak. I don't recommend any kayaks from Pelican. They just don't appear to me to be sturdy. I'd go with a used more "pedigreed" fishing kayak instead, but whatever you get you'll need a kayak.
You'll need a paddle. Big box stores always, always, always recommend paddles much longer than appropriate for most paddlers. A long kayak practically forces you to a wide, sweeping stroke, which makes both speed and tracking difficult. For most recreational kayaks, I use a 205 or 210 cm paddle, whereas many times a big box store will recommend a 240cm paddle. Regardless of what length you get, you'll need a paddle. Carlisle makes some sturdy ones at a reasonable price. In general, lighter is better. I like either fiberglass or aluminum shaft with plastic blades for fishing.
You'll need a life jacket / PFD. I don't care if you're a good swimmer, and I don't care if the lake is always smooth and easy in your opinion. Get and wear a PFD. There's nothing macho about drowning. Get one that is comfortable and that you will wear. If you're going to spend an extra $20 or $30 anywhere in this initial setup, the PFD is a good place to do that if it gets you a more comfortable one that you'll be willing to wear.
Now the fishing part. It all depends on your quarry, but most likely you'll be going after bass if you're lake fishing. My recommendation is a 6'6" Medium spinning rod. There are a million different options for you here. Bass Pro Shops and Cabelas and Gander Mountain all have combinations of rod and reel that are balanced, reliable, and affordable.
If you have a rod and reel combo on your kayak and don't have a way to secure it, you'll lose it. Get a Scotty or RAM mount rod holder and mount it to your kayak. Do not get a Berkeley.
Some lures would be good. I like soft plastics. With a small assortment of terminal tackle and lure shapes and sizes, you can fish practically any condition. The very most versatile combination is a 1/4 oz ball head jig heads and some 3/0 extra wide gap (EWG) hooks for your terminal tackle and a few bags of 4" grubs in purple, black, motor oil, mossy green, white, and chartreuse. With that you can imitate nearly anything in any part of the water column. My favorite lure is a Case Plastics Jacks Worm on a 3/0 EWG hook, unweighted, but I mostly fish shallow rivers. That may not get deep enough for taking lake bass. Also, get some poppers. It's hard to beat poppers for huge fun taking bass from weed lines at dawn or dusk. If your kayak doesn't have deck bungees to hold stuff, put some on.
You'll need something to put your tackle in. I have a small dry bag that I drop a variety of soft plastics in, and a small Plano over/under that I keep terminal tackle in.
You'll need a way to get your kayak to the water. If you live on the shore, drag it over. If you need to travel, some way to strap it to your car or truck will do. There are a lot of options. You don't need anything fancy so long as it will hold the boat firmly.
Good luck. It's a lot of fun and a good way to get some exercise and some peace of mind.
- Big D