Let's take a look at a typical entry form and then go through how to fill it out.
(click on the photo to enlarge)
The first field will be about your horse. It's very important to fill this out completely, especially if a passport number is asked for, so that your results will be attributed correctly.
Following that, you should find a section about the horse's owner. Make sure to fill this out fully, too, because the owner needs to have certain memberships in order for the horse to compete and you don't want to waste the office's time by making them check for themselves. Make sure that you print very clearly and use a fine pen if space is limited!
You should also see a section about your trainer. This is important to fill out because entries will generally be grouped by stable so that when it comes to collecting the horses' numbers and splitting stabling/hay/shavings fees, things can be done quickly and easily. It will also make doing your daily entries easier because shows will often print out a 'trainer add form' that lists all of the horses entered under that trainer.
Next, you should see a section with room for multiple riders (smaller shows sometimes only allow one rider per entry form). It doesn't really matter which rider you put first if you're doing multiples as long as you are consistent with the numbering throughout the form. Be sure to fill out the membership numbers here, too, so that it's clear to the office that your riders are able to compete. Date of birth is not really necessary for those who are not entering age-restricted classes.
There will always be a waiver for you to sign at the end of the entry form; don't forget to sign it!
Different shows have different ways of putting the various classes on the entry form. Some forms will require you to put a mark beside the classes, circle the numbers of the classes that you would like to enter, or write in the class numbers in a blank space. Beside the class, write the number of the rider who will be doing that class (rider 1 or 2 from the rider information section). At some shows, you can write a 'T' if the trainer is riding.
It isn't necessary to add up the fees before sending in the form since the office will do that themselves and you are likely to add on extra fees during the show, anyway. The fee section is not the appropriate place to reserve a stall or a camper spot; that should be done on a separate stall request form.
Now, why should you need to give the horse show your address? It actually benefits you to give it to them because many shows will put that address into a database and then send you their future prize lists by mail. The rider's home town is also sometimes announced at certain shows.
If you aren't sure which classes you would like to compete in, you can leave that section of the entry form blank and it will be accepted for the majority of shows, provided you enter the rest of the information in full. Once you are at the horse show you can enter your classes by filling out an add/scratch form the day before you compete.