Technically, I think 'Exchange' Plans *are* private plans - just marketed slightly differently than you might be used to.
But what I do is: check Consumer's Reports, Better Business Bureau, and the general web for information on the various Insurance companies. Check the newspaper - one of the companies offering via the Exchange in my state recently had monetary difficulties and I think is being subsidized by 'somebody' - state, federal, who knows. Maybe even in the process of going out of business. Basically I'd start by trying to find an insurance company you trust.
Then decide if you *have* to see a specific doctor. If you do, check the plans for your company in one "metal" grouping (Bronze, Silver, etc.) on the Exchange. They're supposed to have a link that tells you what doctors are available with that plan. If you need to be positive, write down the plan name (code, type, whatever you see on the Exchange that would let you feel comfortable you are able to describe the specific plan you're interested in) and call the insurance company to ask.
If you don't have to see a specific doctor, go through basically the same process, focusing on one "metal" grouping and verifying that there's at least one doctor who's accepting new patients in the physical area that you're willing to travel to.
Once you have an idea of a plan that seems to have access to a doctor you're satisfied with, then start focusing on price and details. How much is the copayment between the "metal" tiers? Do they cover situations that you're likely to need? What do they charge for drugs and, if they charge less for mail-order is that an acceptable method for you? Can you handle generics or must you have the brand name stuff - how much will they charge for that? Only you can decide what aspects of the insurance are important to you and your family, so take the time to think about it. Do you have daughters that might get pregnant? Is adoption in your future? How likely are you to need emergency care (got a lot of rough-housing kids? someone with a bad heart? teenage driver who likes to text while driving?

) And before you buy, double check that the doctor you want is in the plan you decided on.
I'd also go to the website for the Insurance company you've decided on and see what options they have available outside of the Exchange. To my knowledge, there's not much reason to go through the exchange if you're not utilizing the subsidies. And the company I had last year definitely did NOT want to have the ACA upheld, to the point that they did a really poor job of implementing Exchange plans into their existing systems. So if you can go through their "normal" channels, you may find life a little easier.
But, basically, your first priority is deciding on the Insurance company you want. The second priority is determining what kinds of services you absolutely must have. From there it's a matter of sifting through the options. And, no, it's not fun. Good luck!