Okay, here's the thing; whether you know it or not, this is a two-part question that you have asked.
1. When will wildflowers be blooming in Austin? As cold as it has been, we are guessing the really good shows of wildflowers in Austin will be from mid-March to mid-summer. We have listed some of the Central Texas wildflowers below, with their projected bloom times.
2. Where is it okay to pick them? There will be wildflowers blooming in the Wildflower Center. There are big signs all over the Center saying NOT to pick them. And, believe me, here in Central Texas we take our wildflowers very seriously. The State Highway Department frowns on picking wildflowers on roadways, both so others can enjoy them and for safety.
Consider: If you take a wildflower bloom, you not only have denied other people the pleasure of seeing them in bloom, but you have cancelled out the seeds for that plant. Many wildflowers, including bluebonnets, are annuals. Pick enough blooms and there will be no wildflowers there next year. The property where it is may belong to someone who raises them to give everyone pleasure. If we publicized (or even knew of) a place where it was okay to pick the wildflowers, they would all be gone before you could get here to see them.
Please - take pictures. Plant wildflowers on your own property and share them with everyone if that is your choice. Please read this previous Mr. Smarty Plants answer on removing plants from property you do not own.
So, if you are going into planting wildflowers on your own, we suggest you read all of the information on the webpages on wildflowers we are linking you to below. Also, read our How-To Articles How to Grow Bluebonnets and Large Scale Wildflower Planting.
Native wildflowers in Central Texas: (we only chose 12, because of limited picture space)
Amblyolepis setigera (Huisache daisy) - annual, blooms yellow, March to June
Asclepias tuberosa (Butterflyweed) - perennial, yellow, orange, May to September
Castilleja indivisa (Entireleaf indian paintbrush) - annual, red, orange, March to May
Coreopsis tinctoria var. tinctoria (Golden tickseed) - annual, yellow, brown, February to November
Echinacea purpurea (Eastern purple coneflower) - perennial, pink, purple, April to September
Gaillardia pulchella (Firewheel) - annual, red, yellow, brown May to August
Lupinus texensis (Texas bluebonnet) - annual, white, blue, March to May
Melampodium leucanthum (Blackfoot daisy), perennial, white, yellow, March to November
Phlox drummondii (Annual phlox) - annual, red, pink, purple, March to June
Ratibida columnifera (Mexican hat) - perennial, orange, yellow, brown, May to October
Tetraneuris scaposa var. scaposa (Four-nerve daisy) - perennial, yelllow, January to December
Wedelia texana (Zexmenia) - perennial, May to November