I've got a flight to DFW (from LHR) on Monday 16th Jan but I need to get to Austin (Downtown Marriott). I chose to go to DFW because I could get BA Club World for ~Ł2000 whereas business class to Austin was ~Ł7000.

Outbound I land in DFW at 3:25pm. I'm debating whether to book on a connecting flight and, if so, at what time. Alternatively I could just get a car service all the way for ~$300. Would that be a better option?

On the way back (on Saturday 21st Jan) I need to connect with a flight departing 3:43pm from DFW (actually via BOS). I'm even more tempted to get a car back becuase it is more reliable than flying. I'd even consider a bus (but it doesn't seem that buses go from Austin straight to the airport). Any thoughts?

FlixBus offers a one-seat ride between Austin and DFW, but the trip takes about 4.5 hours.

I would take a connecting flight. Immigration at DFW usually goes pretty smoothly - especially if you have Global Entry. I'd suggest a flight 2-3 hours after your scheduled arrival. I wouldn't worry too much about reliability of the flights as there are flights DFW-AUS every hour or so. Returning, I'd give yourself about an hour and a half if no checked bags and 2.5 if you're checking bags - bags at DFW are pretty slow and you'd have to re-check on separate tickets.

A car service, I think, would be quite time-consuming given the significant traffic that can often exist between the two cities. Your arrival date is a holiday weekend as well, and in my experience that just means even more traffic. A slightly cheaper option, if you are set on ground transportation, would be the Vonlane luxury bus. You can Uber or take the DART train to the pickup point, which is the Doubletree by Love Field (about 30 mins from DFW). The bus will take you to the Hyatt Regency across the river from downtown Austin, from which you could then make your way to the Marriott. However, I think this would be a bit inconvenient for someone departing DFW.

There are many ways to get between Austin and Dallas....
* Amtrak
* Luxury Coach (Volane and I thought there was supposed to be another)
* Discount bus: Greyhound, Megabus, Flixbus, and perhaps others
* Drive
* Car Service
* Fly

If you will need a car in Austin anyway, you might want to drive. However, I've driven from DFW after a long-haul flight (from DOH), and it was not a pleasant experience. Between the jetlag, the traffic, and me just wanting to be home, I really wished I had booked a flight. It was a solid hour+ of traffic before things started to settle down, and by that point the construction on I-35 took over. At least for you, that Monday is the MLK Holiday, so traffic should be a little less (all government, most bank employees will be off, along with others).

In the downtown area of Austin, parking is at a premium. You might be able to drop off or pick up a rental car there though, and avoid having to store it (just use it for your drive in/out).

Since you'd be doing separate tickets, you'd need to allow plenty of time if you fly. As cactus584 noted, I'd allow at least 2-1/2 to 3 hours to allow for claiming and re-checking.

For $300 for a car service to your door, I'd probably just go with that. They will probably get you to DFW early, but you'll have time to lounge hop (along with the BA lounge you can try the AA Flagship Lounge).

Amtrak is very, very slow, and possibly does not run every day.
If there is another service besides Vonlane, I have not heard but a search did find one. Looks like RedCoach is now operating in Texas (in addition to Florida).
And being a budget traveler, I've tried Greyhound, Megabus, Flixbus, and let's just say it can be less than ideal, Greyhound seems to be always late, sometimes HOURS
Flying still seems best, though intra Texas fares are ridiculously expensive

You should look at Hitch. Its like uber/lyft except for long distances between cities

On January 16, all of AA's DFW-AUS nonstops (except for the 7:00 pm departure) are available for $130 in Basic Economy. But since, according to his profile, the OP is BAEC Silver (oneworld Sapphire), he would still get free checked baggage and a free Main Cabin Extra (extra-leg-room) seat, even on a Basic Economy fare.