I lived in Brenham for many years and my mom still lives there. The flowers vary year to year depending on rain and how late the winter runs but mid March through the end of April is a good window. Most people drive up from Houston and park on the main highway and trample everything to death but you can take some of the back roads and a lot. Hwy 290 is the main highway but a smaller FM 390 is a very good road to take to see some flowers.
Plan to stop at The Antique Rosé Emporium in Independence Texas which is about twelve miles north of Brenham on Hwy 50. This is a retail nursery that specializes and grows on site antique roses (non-hybrid or American roses). Their retail center has grown over the years and includes several buildings like a chapel, a 100 year plus home, and a kitchen from the 1800's and all of these buildings are heavily landscaped with roses and native flowers. If you drive to Brenham to see flowers this place is a must stop.
There are two Blue Bell factories in town and the main plant on the loop is also landscaped nicely with lots of antique roses and flowers. Tours are available but I think it's jut Monday through Friday.
LBJ's wife Lady Bird Johnson is responsible for most of e wildflowers that have been planted with a big campaign she pushed back in the day. I can't tell you exactly where but in the Austin area there is suppose to be a lot of fields planted.