Welcome to Head-Fi!
Before I bought my K712s, I was in the same situation as you. I had only my Astro setup (A40s and Mixamp) and was looking for a serious upgrade. I ended up getting the K712s, a SoundBlaster Omni for my PC gaming and music needs and a Schiit Vali to power the cans.
An amp will make a huge difference when you're using the Astro Mixamp. For me, the Mixamp alonr was too quiet, made the highs harsh and sibilant and just didn't make the headphones sound pleasing. When I added the Vali into the chain, the amp corrected all of those issues.
As for potential PC gaming, the Omni is an awesome device and you may want to consider it. It seems to work decently well with the K712s without a dedicated amp but for me, harsh highs are still a slight issue.
Stillhart will be able to explain further and back me up here. He was one of the people here who helped me with my purchases.
tl;dr Get an amp for the K712. The Vali pairs really well with the AKG K/Q7xx line from what I've heard and experienced.
Edit: I missed the part where you said they'd be for portable use. As Murder Mike said, they're open so they won't work well in a noisy place and may disturb those around you as they will hear what is playing through the headphones. Open headphones won't be a good choice for portables. The K712s are also big cans and not portable.
I personally use a set of Vsonic GR07 Bass Editions for portable use. They cost me $150 and are actually pretty amazing for the price IMO.
Thank you, Thank you, Thank you.
This thread and guide are AWESOME btw. Thank you so much MLE and everyone that continues to contribute.