Austin is a very interesting city, but I would suggest that you consider going during the South by Southwest Festival (SXSW) in March (in 2017, it will be the 10th through the 19th). Austin is, even in non festival time frames, a very music oriented city, but during the SXSW, there are bands playing all over the city!
Sixth Street is a music lovers paradise, and on the weekends, it is pedestrian only. With the U of T just a few blocks away, there are major crowds of college students out having a good time, surfing the bars, and just having fun.
If you like BBQ, you are going to think you have gone to BBQ heaven in Austin! My personal favorite is the Salt Lick BBQ which is a little bit outside of town, but in & around town, you have Stubbs BBQ and there are Rudy's scattered all around town.
If you can't go during March, and it is hot (which is common in the summer there!) try going for a swim in the Barton Springs Pool. It has got to be the coldest pool I have ever been in !
Excuse me for rattling on, you only asked what time was the best and you didn't ask for a list of places to go to, but have fun !
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The best time is when you can. The weather here is variable year round. During winter it can be in the 30's as a high one day and 60-70's the next. Spring is usually our rainiest period but again variable in temp and precipitation. Glorious days can end in rain and cold. Autumn may begin in October or December depending on the year. Summer can serve up coastal humidity on a given day or desert moisture.
This is Austin. It's part of what makes us different (and strange to some). Come and enjoy what the city offers and forget about what's "best".
The only way you'll ever decipher that is from multiple visits and finding what you like best.
-BTW, I moved here is 1985 and could live anywere I want (globally). Austin is my choice. It's that good!
I would recommend from March to May, or October-December. Our winters can have some gorgeous days (today, in fact, it will be sunny and in the 40s) but Jan. and Feb. have unpredictable weather. The summer is pretty brutal starting in June. By mid-July to September it is extremely hot. Though, that is when we find swimming holes and wear flip flops everywhere. Safe travels!
Spring and fall. Summer is oppressively hot and muggy, and winter is off-season...it's not too cold, but the town quiets down and many attractions in the area close or have reduced hours. The bluebonnets and other wildflowers are in bloom in March and April, and UT football season is in full swing in October and November and everyone is having a good time. In spring or fall you can expect high temperatures in the upper 60s to low 80s and lows about 20 degrees below that. You'll have the occasional rainy day and the occasional spring thunderstorm and tornado warnings but the foul weather never lasts for more than a few hours at a time.
In the spring when the bluebonnets are blooming. So beautiful. You definitely want to go before it gets too hot.