Since you are in pre-searing, the party size is only 2, so that makes things easier. But the same things apply to larger parties.
The best way to use the Party search thing is to start by putting a message in ALL text. If you put LFG (looking for group) or LFP (looking for party) in the message it will automatically be put in the search box. It's also best to supply a bit of information in the message, like, for example, "Warrior LFG mission" or "Ele LFP for SomeQuest".
One reason for starting by putting a message in ALL chat is that some people still don't know that the Search box exists and they don't look there. (noobs!

)
Once you've entered the message, it will also appear in the Search box. If people want to join your group (join you), they will click on your name in the Search box and hit "Join". This will put a symbol next to your message in the box and also put something like "<playername> has sent you a party request" (I don't recall the exact message) in the text box. Then you can just accept them as usual.
If you are trying to join someone else, then you can click on them in the Party box and select "Join" and they will get the request messagse. You will join their party if/when they accept - even changing districts automatically if needed.
Here's a handy tip for those people who don't use Search, or have not worded their message in such a way as to make it appear in the Search box:
Click on their name in the chat text window. This will put their name in your "whisper" box. It should already be highlighted, but if not, highlight their name (click and drag over it.) Once it's highlighted, hit Ctrl-C (copy) to copy the name. Then click on the "add player" box in the Party window and hit Ctrl-V (paste) - it's not actually called "add player" btw. Then hit the little "+" symbol by their name to ask them to join you (or you to join them).