Over summer I'm going to be visiting relatives in Dallas, and plan on visiting Austin, Houston, and San Antonio too. I don't think I can afford a hire car. I've been told though that not having a car will be a problem. What is the public transport like between these cities, and in the city centres themselves?
In San Antonio most of the major attractions are within walking distance of each other; so if you can get accommodation downtown you should be fine.
Austin has decent public transport (for Texas) and if you stay close to downtown you shouldn't have a problem.
Houston is going to be your biggest problem, you will HAVE to hire a car, or skip it. It's huge, spread out, and has sparse public transportation. I would say, unless you have your heart set on seeing something in Houston, skip it and spend more time in San Antonio and Austin, and/or add Fort Worth to the itinerary.
Dallas to Austin and then on to San Antonio your two options are Amtrak and Greyhound, both fairly inexpensive, especially if you buy your tickets in advance. Go to their websites to check out prices.
To Houston the only option is Greyhound, and the Houston Greyhound station is not in a location that is in anyway convenient for a tourist. The city is so spread out that taking a taxi anywhere would be $$$$$. Yet another reason to skip Houston. Check out Fort Worth instead, if you weren't already planning on it. The downtown is compact, walkable, and there is stuff to do and see. The Amtrak station is located there, so if you take the train you'll have to go to Fort Worth anyway.
I hope this helps.
One more thing...just for your information: There are Mexican bus lines that go from Dallas to Monterrey for a very inexpensive fare. The bus leaves in the evening and arrives in Monterrey in the morning. In Monterrey, Barrio Antiguo is supposed to be really cool and fun, with lots of bars and cafes. So, if you have time and you are allowed to pop over to Mexico with your visa, it's a really fun thing to do.
I assume that public transport could include airlines.
Southwest has excellent schedules (and fares) between all you mention. Some Amtrak. Good intracity in Dallas (DART), the TREE is local rail between Dalls and Fort Worth. City buses in Fort Worth. Don't know about other towns, except Houston is REEEEEaly SPREEEEEAAAAD out. If you like art, FTW has several internationally reknown art museums.
Check prices thoroughly. A car hire may be your better deal. Also, check your maps. It is four hours driving from Dallas to Houston (straight interstate, no stopping) and three hours Houston to San Antonio, or Dallas to Austin. If you are summering, is there a "network" you can use of friends and family for places to go? Get back with me about more information. I was born in Galveston, I have lived in the pine woods of East Texas for seventeen years, and I have burned a lot of gasoline on the Texas roads!
houston has convenient transit for 1 very specific area. most houstonians call it montrose. the area is bounded on the north by allen parkway, to the east by downtown, to the south by the southwest freeway and to the west the galleria (the galleria is extreme west of monstrose and not usually referred as bieng in the montrose area). The reason why this are is convenient is because the 82 bus line runs every 12 minutes at every other corner on westheimer.
this is good because westheimer is right down montrose east to west. the bus time from downtown houston westheimer is approx 20 minutes and all buses are air condition. the 82 is usually packed during commute to work times, most of the other time the bus is empty. also, from westhiemer to the north or south, you can reach both either end with less than a 10 minute to 15 minute walk. be warned houston is hot through end of october so shorts are a good idea as houston gets nasty hot.
carry a bag with wet towels in a sandwich bag to freshen up and duck into any shop to cool in the air condition. most shops are along westheimer between taft and s. shepperd. however s. shepperd has many shops as well. along w. alabama and richmond also have areas of shops, and w. gray too.
all streets mentioned have regular bus routes of about 15 to 20 minutes, and bus trips are quick because traveling distance is short. the exception being the galleria, but the galleria is a must see.
other must sees in houston is herman park area (south of montrose and main -- approx 5 minute walk).
if you are not staying in this area, hire a car.