The kind of therapy you should get depends on your issues as well as your preferences. There is a vast number of different therapies out there. The most popular ones would probably be: psychodynamic, CBT, DBT, maybe also schema therapy. There's other forms of course, but those are the ones that come to mind for me first.

Usually CBT and psychodynamic are a good start for people (though you will find lots of people with negative views on CBT on this forum I think). Keep in mind that lots of people have to go through a few different therapists before they find somebody that they feel comfortable with. It's best to just go to a session, see how you like the therapist as well as their style of therapy and if it doesn't work for you, search somebody new after a session or two.

You usually do not already need to know what your diagnosis is before seeking treatment. It's part of the therapists' job to figure out what your issues are. I can't tell you whether insurance will cover it if you do not meet the criteria for any diagnosis. It depends on your specific insurance.
And as for weekends, some therapists I imagine work on Saturdays or even Sundays (though keep in mind that being a therapist is a normal job and people usually work Monday through Friday). But as with any services offered on the weekends, it might be harder to get a time slot.