I have worked in two law firms so I will try to give you some insight.
When you first arrange to meet firm X or Y they will send you a couple of their partners. Trust me when I say you will probably never see these people again unless you try to leave the company you are going for.
At the initial meeting listen to all the information they have to say. Ask them about rates, any similar companies to you, what extras they have to offer (sometimes they will offer you massively reduced rates on other legal matters (moving home, wills, etc)).
After this meeting go away think about what they have said. If you liked the partner arrange meeting two. Arrange this in a local coffee shop. Say you are really interested in going with them but you would like to meet the persons handling your case load. Most firms will be more than happy to do this, (some will cringe).
At meeting two if the person there is not a junior case handler then know they have not sent the person handling your case load. Lawfirms (especially big ones) will not hand a small account to a partner or senior case handler. Again ask questions and see if you get a long with them.
After working in a law firm for a couple of years I can tell you there are two types of case handlers. People you will get along with (these people dont generally have there head up there arse) and people you wont (these people generally think they have the biggest go-nads in the room).