The best way to approach this would be to have a master plan developed with instructions on how to break it into sections that can be done DIY maybe twice a year, each part being within your stated budget. I used to do plans like that with each section at about $500 or so with instructions on what must be done first and what should be done in spring vs fall. Generally in 4 or 5 steps you can get all the parts put in place, taking a year or two to complete it. You want to add trees if needed very soon, since they take the longest to fill in, but a lot of things need to be planned before you do that.
The very first things to consider are water and hardscaping. Water means both water in and water out, so you must consider irrigation and drainage, first and continuously with each change you make. Then look to the permanent things like patios, arbors, paths and such. I suggest getting the plot plan of the house from closing and blowing it up to a scale of 1" = 10' or 8' or 4'. Mark distances on it where needed. Then use tracing paper and designate special areas and considerations; noisy pool equipment, neighbors windows, pathway for the lawnmower, tool storage, trash cans, etc. Put arrows on the paths to and from wherever. Consider if paths must be grass or something else, and if that is right away or over time. A grass path can be paved later as needed. Find where the sun and shade are during the year, which way is north, which is southwest where the hot summer sun comes from.
Evaluate where you need shade and how best to provide that, umbrellas, arbors, trees, large shrubs. Evaluate need for screening for privacy or noise control. Last, look at the views you want to enhance or to hide. See the pool from the house and the house from the pool. Ideally, each window from the house has a picturesque view that brings the outdoors in while allowing safe viewing of kids or privacy where needed.
Here in Houston, there are two planting seasons, spring and fall. Trees and shrubs are best planted in fall here, though Dallas may be different. Prices are better in fall as the nurseries go to 70% off to clear room for Christmas trees, but selection is worse. Selection is better in spring but prices are high so get the "special" ornamental things then.
Finally, not at the start but at the end, look at specific plant materials. There are lots of choices for various needs and it is easy to substitute when you know that you want "a large evergreen shrub for screening that is 5' tall at maturity" or "a medium sized evergreen tree with seasonal interest with small leaves" or "a small flowering shrub that will not mind some salt water and hot sun". The local nursery can steer you toward the plants that will tolerate you soil and heat/cold and still fill those needs.
Keeping the irrigation system current when the plan is evolving is a challenge and it may be that you will lay out the system and the beds a year or more before they are all planted. It is OK to lay the irrigation, bring in the soil and then just keep it mulched until the correct season. And you can use temporary things to fill gaps as you wait for the permanent shrubs to fill in.
I don't know much about Dallas but the soil there is tricky. Look at ornamental grasses used in commercial areas, inexpensive and beautiful. If you go to the arboretum, you will see a lot of azaleas that really do not do well there without massive soil work so beware of that. Better to visit the Lady Bird Johnson Wildflower site for inspiration.
