Utah is great in April! That is the ideal month and October on the other end is also nice. Mid April is when I did my longest (3 week) trip there. March could be a tiny bit iffy depending on the weather.
I would guess you might want to fly.
Canyonlands is my favorite, though the whole state, especially in the south with its 5 national parks, is fantastic. And the NPs are only the tip of the iceberg! You have to be very methodical about water sources and such. Also Utah can be very windy, but in the spring and fall it can be windy as hell. On the other hand it is more remote. If you go for a week or more you will really be alone.
Another twist you can think about is going down the Green river and hiking out. You can rent a canoe in Moab, and they will shuttle you up river. You then float down through incredibly beautiful canyons (think grand canyon beautiful) about 100 miles. At the bottom, just below the confluence with the Colorado they come by air boat from Moab via the Colorado and pick up the canoe. You can hike out or take the boat back to Moab. Less actual hiking the first week, though there are miles and miles of side trek you cab make on the way down the river, with side canyons, remote Indian ruins, and so on. This book:
is a fantastic guide if you want to know what is there. Ton of information on history, routes, water sources, geology, and best of all tons of detailed hand drawn maps. His other book on the area are also killer.
Doing that will be a little less lonely (in the danger sense) since on the river you will see other boater a few time a day at least, but otherwise you will be alone.
There are load of really life changing places to go in Utah, but I would say a 10 day trip in many of them requires a lot more self sufficiency and route finding than, or example, ANY place in the Sierra.
A really good one with 3 or 4 incredible arches - almost better that anything in Arches NP, and one arch that is simply monumental, is Coyote Canyon in the Escalate System. Most of the popular canyons will have running water, so that is nice too. I would almost say that Coyote would be the ideal hike, out and back, for a solo hiker, first time in Utah, with pooch, for a week or more, except for the really difficult logistics. You would have enough people around not to feel totally isolated, but not enough to feel like they were impinging. But, the logistics! The town of Escalante is out in the middle of nowhere, and it is a 90 minute drive from there on a dirt road to get to the start. Some people here might know of some way to get a shuttle, but getting to Escalante might be difficult without renting a car. If you don't mind renting a car for a week just for the shuttle then you could fly to Salt Lake City.
Grand Canyon will also be ok then. Maybe snow on the south rim (access to the North rim might still be iffy), but nice down inside.
The closest thing to Austin for an epic adventure, at least how I would define that, is Big Bend NP. You *have* to go there some time while you are in Austin. Still a long drive - more than a day probably. I once had a week long trip planed for Organ Pipe NM in southern Arizona, and the ranger I talked to on the phone was not encouraging. He said that drug runners from Mexico went through the back country all the time and was worried they might go all "breaking Bad" on me if I ran into them. LOL So I changed my mind. You might want to call them at Big Bend NP and see what they think on that issue.