The same kinds of basic that apply to any facility:
Keep bathrooms clean. You might be surprised how many customers are turned off by smelly or poorly working bathrooms.
Keep floors clean, so you can do exercises without bringing your own mat. (Personally, I don't worry that much. I've done a lot of camping, kayaking, etc., which involve playing in dirt and mud. But lots of people are picky.)
Maintain rental equipment in good condition. E.g., sharp blades, reasonable stiffness. Make them available at least 30 minutes prior to session.
Make sure someone checks that people go on the ice with properly tied skates. I believe that failure to do this is the most common cause of people doing poorly, and of a large fraction of injuries.
Have a phone number. Have someone answer it.
Same with email.
I agree that web pages should be viewable on mobile devices. Keep the web page simple - just basic HTML. Remember that some screens are very, very small, so don't try to force a specific layout. Don't assume helper software, or specific platforms. Make sure everything is bookmarkable, including session schedules. You don't need a separate mobile site - this should all apply to your regular page. (And I agree that timely posting of schedule changes is crucial.)
Most of all, keep to published on-line and paper schedules. I hate going to a promised session, and discovering that someone rented the time spot. I hate discovering that the session ends earlier than promised. I hate telling a customer that these promises have been broken. A lot of other things can be forgiven, but there is almost nothing that annoys and angers customers more than breaking your promises to them.
These are all quite basic. Except for the rental equipment, and properly tied skates, everything I've mentioned should be expected of any good business or facility open to the public. Yet, I know of no rink that meets them all.
P.S. I know that not all these have to be done by the managers, so it goes a little outside the parameters the o.p. set. For example, someone else can check and clean the bathrooms. But good managers need to make sure they
are all done, on a routine basis.
P.P.S. Have any of you ever wondered how facilities that costs millions of dollars to build, and have operating budgets of a million or more dollars/year, can make such basic errors? So have I.