Which press does the serious reloaded choose?

Dillon Super 1050

But I bet you are asking about the first press a novice reloader should get:: in that case ANY of the typical presses are just fine.

For precision  stuff Co-ax.  Lots of people do case prep on a progressive.

Depends on what you mean. If you're asking the question then you don't reload what you mean by serious is probably flawed.

So, a CH4 4 station is a nice multi-station non-progressive used by competition shooters. My coach and I both use one, so do most of the guys we compete against. They're great for taking to a range if you're dong a bench match or you have another good reason to be handloading at the range. https://www.ch4d.com/products/equipment/presses/044000 These are super easy to use and set up. They're stupid strong and durable. Expensive but worth it especially if you need something between a single stage and a progressive press but you don't like turret arrangements or you want to take it to the range. It's really fast to use like a prog. If you have time and money this is a good way to go.

RCBS Rockchucker single stage is the old standby for newbs that want to not cheap out and want good kit for a lifetime. http://rcbs.com/Products/Presses-and-Kits/Single-Stage/Rock-Chucker-Supreme-Press.aspx The simplest and one of the toughest there is. Everyone I know has at least one Rockchucker. If you have time but not money this is the way to go. I have one of these which I use for load development.

Dillon 550 or another progressive press is what most hardcore shooters will eventually upgrade to if they shoot high volume enough or want to save time enough. http://www.dillonprecision.com/rl-550c_8_1_23594.html Progressive press, very difficult to set up compared to single stage or turret. Once you get into a dillon you'll go broke on accessories. If you have money but completely lack time this is the way to go. I have one of these that I use for all of my high volume ammo production, especially match rifle ammo.

Lee Challenger Anniversary Kit: If you don't have money (the kit is about 100 bucks) and you have plenty of spare time and neither of those situations are likely to change anytime soon, this is a great way to get into reloading without making a huge investment. It's sufficient to turn out great ammo but is not nearly as sturdy or slick as more expensive options. They're plenty durable. I've used my roughly for ~20 years and it's still chugging along. http://rd.bizrate.com/rd?t=https://ads.midwayusa.com/product/423081/lee-challenger-breech-lock-single-stage-press-anniversary-kit?cm_mmc=pf_ci_connexity-_-Reloading+-+Metallic+Reloading+Presses-_-Lee-_-423081&mid=77064&cat_id=12150125&atom=10468&prod_id=&oid=6045349214&pos=1&b_id=18&bid_type=10&bamt=cfbf9532e1e82faa&cobrand=1&ppr=b7affea52df7762b&rf=af1&af_assettype_id=12&af_creative_id=2973&af_id=26865&af_placement_id=1659178&dv=bc8b5aa76eb237768d786e8a48e2cde3&af_campaign_id=j2ndevbf4100zk8a03b2f

There's also the Dillon 650 and Super 1050 which are things you'll get if you know you need it. If you don't know you need one, you don't.

Those that are Al Gore level serious would probably get a Camdex machine because why pussy out when it comes to free shit to drink. http://www.camdexloader.com/

The press doesn't matter. Lee will work

Someone please tell me how a CO-AX makes better ammo than any other press, you cannot say straight seating because I have already tested this.

With a Forster Ultra or Redding Comp seater my 18 year old, sloppy auto progressive press makes straight consistent ammo.
Outside of seating a bullet straight and not messing up the brass(more related to die and brass), the press you choose has almost nothing to do with the quality of ammo you make.

The ergonomics of the Co-Ax suck for speed as well.

Disclaimer: I haven't done any reloading in years (but may be getting back into it for precision rifle ammo)...

I don't know that the press is all that important as long as it is functioning correctly...  Dies are another question entirely and I am sure that there has been many a long discussion/debate regarding reloading dies and which is/are best...

I think that a good single stage press is all that you really want initially as you are more interested in quality of the final product than volume/speed of loading.  RCBS, Lee, Hornady all make good single stage presses.  When I was in my late teens and early 20s, I loaded more stuff on an RCBS Rock Chucker than I tell you.  Great press and unkillable!  

Best of luck.

Period. For all the money I have in presses, my 20 year old lee turns out just as high quality of ammunition as my RCBS. Honestly, I think the more complicated your press the more likely you are to make mistakes that result in crappy loads.