First, one should note that bei is in general not used with verbs such like gehen, kommen or other words which would express an idea of movement. One would say:

Ich bin beim Arzt.

but:

Ich gehe zum Arzt.

Now for a few guidelines about the other prepositions:

Nach is used when you go to a city, a country, or any other named inhabited settlement of region.

Wir haben uns entschieden, nach Frankreich zu fahren.

Zu is when your destination is a building.

Ich muss für dich zum Rathaus gehen, damit du dein Zertifikat endlich hast.

In is used when the idea to be in the place is important. A few examples, from the top of my head: "ins Meer", because you will swim in it, "ins Kino" because you want to watch a film in it, "ins Büro/ins Geschäft" because you go there to work. Note that the use of in and zu are very similar, and you nearly have to memorize which preposition to use in each case.

The use of an is quite difficult to define. I would say the most common use is when you want to go to some kind of border. 'See' and 'Küste' can be understood in this case as a border between land and water. This can be used naturally in the case you go there, but also when you are there.

An der Nordseeküste, am plattdeutschen Strand, ...

The other case I found is a figurative one:

An die Arbeit gehen : to get to work, to go down to business

To sum it up, those are guidelines, and unfortunately, in most of the cases, knowing which preposition to use gets only easier with how familiar you are with the language.