practically speaking, a barrier of some sort would keep most ground covers from extending into your lawn. i have some of those interlocking, pound-in scalloped plastic pieces around a bed and it works very well and was easy to install. you hardly notice it during summer.
as for plants, two that i grow are golden creeping jenny, lysimachia nummularia 'aurea'(needs some sun to maintain the good yellow color; very low growing and easy to pull out of unwanted areas) and ajuga reptans 'burgandy glow'(variegated, for shade).

Pachysandra is the queen of ground covers! Plant it fairly thickly, it is not very speedy about spreading, but it looks great! You are wise to get the ivy out--that stuff is horrible!

There are a few types of sedum that are used for a ground cover. Low growing, non invasive, and can take Texas droughts as it's a succulent.

Verbena is one of my favorites in full sun.

Ophiopogon, commonly called monkey grass (it is not Liriope), is the BEST ground cover for my money. It does equally well in full sun or dense shade, is drought resistant, is thick enough to prevent most weeds, is not a maintenance nightmare, and just plain looks nice. Do a google search for Ophiopogon to see how outstanding this ground cover is.

I like Ceratostigma plumbaginoides -it spreads slowly, grows in sun or shade and tolerates drought. It can be propagated by cuttings and by scattering the seeds that are produced in the fall. It is not evergreen but has beautiful, incredibly blue star-shaped flowers from mid-summer to fall. In addition, the leaves turn orange in the fall.

Thank you so much, Susan! I have used that plant and love it, but I never knew the botanical name for it. I call it plumbago, but that confuses people who are thinking of the large shrub.
You are right, it is tough as nails and the color is great. Cheers!

Helianthus debilis....beach sunflower. Very easy and not especially invasive.

E-mail me if you would like some cuttings.