Well, it depends on the level and type of competition. There's no short answer to this question and no one experience. Every single competition is different.
What are different types of competitions like?
There's really no standard. But if it's a local competition, you go to the venue, sign in at registration 2-3 hours before you skate, and are given a change room number. You warm up, change, then go on the ice for a timed warm up. Then you compete.
what are overnight competitions like?
Again, they vary. But you'd stay in a hotel (either by yourself, with teammates, or with family), and return to the competition venue the next day. You usually have to sign in (and sometimes out) for each event.
How do out of town competitions usually work for you?
Well, my out of town competitions tended to be across the country (when I was skating I lived in a major city, where 90% of provincial and national competitions were held) or in Europe, a bit different from anything you'd be doing. But it depends if you're driving, flying, staying overnight, etc. Basically make sure you've packed everything (make a list) and plan to get to the competition in good time (look at traffic and weather reports; I've seen dozens of people miss their skate because of that).
does your coach come with you to competitions?
Mine always did. Most coaches charge a fee for out of town competitions (which can sometimes be split between multiple skaters).
What are inhouse competitions like?
As in… your home club? Way too complicated to answer. Literally every single one is different. Mine is just a standard competition.
what is a standard schedule for leading up to a competition (eating, sleeping, getting ready, etc.)?
Varies by person. I never had a special schedule. I always packed the night before, and made a list so I wouldn't forget anything crucial. Getting enough sleep is a good thing to do in the week leading up, as well as eating well (no chips or chocolate - lots of protein and fruits and veggies) and staying hydrated. But this was just a habit for me anyway, nothing I had to think about.