I was going to had to REI and look at Kayaks but don't trust myself with just buying anything yet.
What kind of gear would I need to lake kayak? I am assuming the boat, paddle but I am sure there are other items.
I'd carry a whistle just in case and a cheap pump to pump out water if you get swamped without overturning. Make sure you have a hat and some sunscreen....sunburn happens! Bug repellent is good if you are on a buggy lake. I am a lake only person and have found these things useful. The pump is good if you paddle on a lake that has waves generated either by PWC or wind. Waves can swamp you pretty quickly and a pump is valuable to have in this situation. Of course a life jacket is really necessary.
In Vermont it is mandatory.
If you use a sit in boat a spray skirt is great for keeping the water where it belongs.
As mentioned dry storage for phone, camera, towels and hey ya might want a picky nick so throw that soft side cooler in the boat too.
Water chap stick and sunscreen can be worth their weight in gold. Its unbelievable how thirsty paddling can be. I forgot my chap stick last time and after 3 days my lips were raw.
And the recommendation for a proper kayak PFD is spot on. It needs to have a higher rear bottom to ride above the back band on your seat. You can imagine how uncomfortable and frustrating a standard boat PFD can get after an hour or so.