I have a trip planned to Austin at the end of May. We are staying right downtown. It seems like there is a Metro Rail to use for public transportation or taxis but no Uber or Lyft. We are not afraid to walk around most of the day. Does anyone suggest renting a car or think it is necessary to see the city?
Walk and Cap Metro ( Bus) are good options. Lots of connections downtown . Also goes South of the river to Clubs & Shopping. Music Antone's, Continental Club, Saxon Pub( South Lamar). On nice weather days nice hike and bike trail around Lady Bird Lake. Near trail end is Chuy's Restaurant good new style TexMex . traditional Mex- Matt's Famous El Rancho ( since 1957) on South Lamar. Franklin Barbecue in East Austin is good , but get there early & check online too. Near Downtown Iron Works BBQ quite good...Also trailer dinning has many options now. Right now teams are rolling soon TXRD Roller derby ( Thunderdome just south of downtown) fun to catch bouts. 1 or 2 a month. $12-1$7 a ticket
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There is Uber and Lyft in Austin. They were out for a while but came back several months ago. And there are other taxi services. If you are staying within the downtown, then walking around is doable.
Uber and Lyft are in Austin, so no issues! MetroRail is actually pretty limited. Bus is decent. You can also take a pedicab, which is a great way to experience downtown. You don't need to rent a car.