Absolutely you should watch it. Your comparison is apt; you're missing most of it if you're just reading the script. The sights and sounds of a good performance make it much easier to follow.

Reading the script is an advanced skill. I think we do a real disservice throwing unedited scripts at students.

As for what to watch, there are many great film versions. Branagh's Henry V and Much Ado About Nothing were my gateway. There is a stunning Titus by Julie Taymor, especially if you like horror -- it's an obscure play but Taymor makes a spectacle of it. There have been several very good Macbeth adaptations recently; I need to catch Denzel Washington's because I've heard it's great. There was also a very good Coriolanus recently (another obscure play given a very modern and relevant performance).

I would watch a film, and if you love it, read the play to find out how it works.