First off, welcome to "Wild Texas"! Hope we can be of some help as you enter the world of camping!
In answer to your question I must say that it depends alot on how far you want to go and in which dierection. If you want to go north about 45 minutes to an hour, then as "Alchemist2000" suggested, Ray Roberts is the place to go. You have two parks there to choose from; Isle du Bois Unit at the south end of the lake and the Johnson Branch Unit at the north. Can't really say which I like more, both being very nice.
From the site "Alchemist2000" linked to, you can access info on all Texas State Parks. Another beutiful park not to far from Dallas is Purtis Creek State Park. Lots of great camping sites there and a wonderful lake to fish and swim in. Guess it's about an hour southeast of Dallas near Eustace, Texas. Have a site:
Purti Creek State Park
Another outstanding SP not too far from you is Tyler State Park. Park is north of the city in the middle of a pine wood. It also has a beutiful lake that you can't help but like. Have a site:
Tyler State Park
Cedar Hill State Park is on Joe Pool and very close to you. There is nothing wrong with it but I don't care much for it as I find it to be too close to Dallas. Just personal preference. Have a site:
Cedar Hill State Park
I have not taken you west of Dallas. There are a number of great SP's in that direction as well but felt that the extra distance involved may not suit some one new to camping. If interested, let me know and I can give you information about SP's in that direction.
Well, that will give you four options to start with. Find camping in Texas SP's to be great and have been doing it for many years. I will warn you that if you don't have reservations, getting a weekend site in most parks for most of the year is difficult to impossable. Spring, summer and fall, these parks book up on the weekends months in advance. This holds true for most all camping areas, public or private. This is why DW and I do most of our camping during the week. You can try calling the individual state parks directly. If they have a site available, with or with out utilities, you can reserve it that way. It will at least save you a fruitless trip if there isn't anything.
Hope I've helped some and you have a great time no matter where you end up! Be sure to let us know how the first trip went. Again, welcome to the group and happy camping!
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Turn Key, DW and Pepsi & Cherry, The Camping Boston Terriers
'03 Chevy 2500HD, 4X4, X-Cab, Long Bed
'04 K-Z "Durango", 275RK ("Sunday Haus II")
Twin Kayaks, "The Ride" by Wilderness Systems
North Central Texas, Where The West Begins!