Cats will definitely use your garden beds as a litter box if you let them. Put in pointy sticks or cover your soil with chicken wire to keep them out.

How much to water depends on your climate and your plants. Don't water a tiny bit every day! You want the water to soak down deep and then let your plants send roots out in search of the water. So get in the habit of sticking your finger into the soil and don't water until it starts feeling dry a few inches into the soil, then water them deeply. (If you were growing seeds, you'd need to keep those top couple inches wet, but with plants you want to encourage them to send down roots).

As for bugs, there are so many types and so many ways to approach them, you're best off waiting til you see something specific and then asking here (or use the search function for your particular plant - this forum is full of years of knowledge).

Good luck, and welcome to gardening!

So, more info would be great...
How big is your new garden?
What kind of format (rows, raised beds, square foot garden, etc).
In GA I would assume you have red clay...but you may be one of the fortunate ones with topsoil or amended soils...so what kind of soil?
Are you planning to use organic methods?

Anyway, good luck with your garden; enjoy the fresh air and sunshine and look at the garden every day. Take pictures or keep a journal to help you in learning and planning for next year. I've been gardening here for 14 years now, and the soil is finally looking like DIRT instead of red clay...but that's only with adding organic matter every every every year.

I am new this year too! I planted the week of May 25th. I have had lots of trials and errors but now the garden is beginning to look great. I will list what I learned that match your plants.
Some problems I had: Watch for baby grasshoppers (I killed 50 in 3 days), aphids, fea beetles, cut worms(those are my pests so far). Flea beetles I find 1 or 2 everytime I check my tomatoes. Zucchini stems split but my plants look great. I thought most of them died from neighbors spray so I replanted seed next to each and will soon thin those out. I transplanted my cukes too early before the plants had more than the first 2 leaves and they died so I replanted directly with seed. My peppers had ALOT of rain. I started them from seed in the house, transplanted them, then we got lots of rain. I saved the best extra plants and was able to replant them for the ones that looked bad in the garden. My toms got tons of rain too and got really strange looking here and there. They have mostly overcome it. When I read about it, I found its probably stress, sunscald and over watering from the rain.
Tips:
Be sure to cover the toms after planting deeply to protect them from sunscald. I just used a couple of plastic grocery bags and cut the handles and sides, then tied onto the cages. They got lots of breeze and less sun...we get 12 hours of direct sun.
I am going to do the drip bottle to water them with milk jugs or soda bottles and use them above ground. (google in-ground watering with milk jugs)
Learn how to correctly water. I am still learning (you gotta use mulch)
Check plants everyday for bugs and bites out of your leaves.
Use mulch!!! I used straw this year, if yours is moldy put it into the compost pile.
Start a compost pile! Its fun!!!
I bought diatomaceous earth to put in my yard to keep pests out. I am waiting for this last rain to stop. We have 2 acres of weedy field/yard and lots of grasshoppers.
I am still learning and this is a great place!!!