When I'm in a party with a friend of mine, how can I follow him? Is there a combination of keys which allows me to run towards him?
And besides that, if one of the party members walks through a portal, does the rest of the party get zoned as well? If not, what's the use of the party then when people are splitted up between zones?
It sounds like you're coming from Guild Wars 1, where the behavior you're asking about was indeed implemented. However, these changes have not been carried over to Guild Wars 2.
No, there is no way to auto-follow someone in GW2. The developers have stated that this is to help discourage botting and also to keep combat and moving through the game world engaging for players. I can't currently find a quote for this, but I do remember this being mentioned in interviews.
If you walk through a portal, only your character is zoned, not the entire party, unlike in GW1 (which made running people very easy!).
The benefits of being in a party are as follows:
You can chat with each other using the "Party" channel.
You can hold Shift+LMB in order to draw on the mini-map for your party members to see, or just Shift+clickLMBinstead of hold, which will create a red ping.
Similarly, you can use Alt+LMB to create a Personal Waypoint on the mini-map or World Map that your party members can see.
You can see other party members as blue dots on the mini-map while you are in the same zone, though this is not as useful if your party is split across multiple zones.
You can rightclick teammates and choose the "Join Instance" option to join them in their instance of a zone/overflow. If you're not in a party with someone and want to join them in a particular instance, it can be very difficult (or pretty much impossible, during large events) because you'll be placed in random instances with no guarantee of whether or not you'll be in the same zone.
Because you get XP for a mob when you tag it whether you're in a party or not, the main benefits of being in a party with people are being able to find them easily on the mini-map and use the above tools when you're in the same zone, and being able to chat with them through Party chat or join their instance when you're not.
In addition, parties will survive character switching and (most of the time) logging out of the game completely. You can switch from your Necromancer to your Mesmer and still be in the same party you were in with your Necro. If you log out completely (not just back to the login screen), parties will still often carry over, but this is dependent on whether or not there was a patch/update in between the last time you logged on or not.
It is rumored that drops when you are partied are better, but this has not been confirmed by ArenaNet. Evidence for this is mostly anecdotal and speculative.
