This is a very popular question. Please read this previous Smarty Plants question from somone planning a trip from Colorado, which will save us some retyping. Then look at this USDA Plant Profile map showing you the counties in Texas where Lupinus texensis (Texas bluebonnet) normally grow. There are some in North Texas, a few in the Big Bend area, several East Texas counties that have them, but you can see that the bulk of the counties are in a cluster that basically contitutes Central Texas and South Texas.
In Austin, of course, you can visit the Lady Bird Johnson Wildflower Center. Follow some of the links on that site to find out what is blooming and when. You will see much more than bluebonnets there, and learn about the many programs and projects we have to preserve the environment with its natural beauty. From Austin you can go west to Fredericksburg, and east to Brenham. Or, coming from Arkansas, you can begin in Sherman-Denison, in Grayson County, work your way down through McLennan County, then to Travis County, where Austin is and then follow the directions above, or go on south to Webb or Hidalgo Counties. Can we guarantee dates and abundance of blooming for any of those places? No. The generally issued forecast is about six weeks of bloom from early March to mid-April. They will begin blooming earliest in South Texas and then proceed northward until they hit the Texas border with Oklahoma.
But, wherever you go, don't just look at bluebonnets. Texas in Spring is a marvel of color and variety, all over.