1) Accept that you're going to pay more upfront. Many insurance providers offer different levels of coverage, using language such as "silver, gold, or platinum" or "basic or enhanced." The way these typically work is that at the lowest level, you pay a lot less per month (your monthly premium) but pay more per appointment, prescription, test, or procedure; these are typically great plans for people who don't expect to have many medical issues in a given year. At the highest level, your monthly premium is higher but your co-pays are lower. If you have a chronic medical condition (notice I didn't say "illness" - I feel that being "chronically ill" is different than having a chronic medical condition), you're probably better off going with a higher-level plan. Although I haven't had any hospital stays for more than a decade and am on relatively few medications, I still go for the highest level that is financially feasible for me. In the long run, it should save you money, especially if something happens. The difference between a procedure being covered at 80% and it being covered at 90% could be thousands of dollars.
2) If you have a preferred doctor, call him/her first to see not only if the office accepts the plan(s) you are considering, but also if the office has a preferred provider. For example, my doctor's office has previously stated that "so-and-so insurance" is a nightmare to work with. If the insurance company doesn't treat your doctor's office well, do you think that company will do any better when it comes to you? Remember that this can vary by region, so just because your cousin's best friend's sister-in-law's neighbor had a great experience with Insurance Company X doesn't mean you will.
3) Check the insurance company's prescription drug formulary to see if your medications are on it. (Be aware that your insurance company might use a pharmacy benefit management (PBM) system such as Express Scripts.) Familiarize yourself with what the different tiers mean and what tier each of your prescriptions is on. Use this information to estimate your co-pays. Medications without comparable generics will typically be of a higher tier and therefore cost a higher co-pay regardless of the plan you choose; however, you may find that if you are on a rare drug, it does not even appear on some formularies. If you are dead set on getting a plan without a necessary medication on the formulary, call before signing up and ask how such a situation is typically handled and what the common work-arounds are.
These are my top three tips, but I'm sure there are more out there! Comment below if you have a strategy for choosing an insurance plan that has worked for you.