should I learn measure theory before learning probability?

The new book on measure theory that I am writing may be useful to you. It's title is Measure, Integration & Real Analysis. The first eight chapters are currently freely available on the book's website: http://measure.axler.net/. More chapters will be available on the website as they are completed.

In fact, it's the inverse. Try some introductory probability books (e.g. Kai Lai Chung's introductory probability book), before beginning real analysis. In that way, you know the motivation for studying abstract integration. If you want an introductory book with more discussions on measure theory, try David Pollard's A User's Guide to Measure Theoretic Probability.

Quoting Rick Durrett from his book Probability: Theory and Examples, "Probability theory has a right and a left hand. On the left is the rigorous foundational work using the tools of measure theory. The right hand 'thinks probabilistically', reduces problems to gambling situations, coin-tossing, and motions of a physical particle."

A lot of probabilistic principles can be learned from finite or countable sample spaces, for which essentially no measure theory is required. Ross's a First Course in Probability can be profitably read without any measure theory. Once you start learning about things like Brownian motion, you'll find that measure theory becomes unavoidable to define the concept precisely. But even there, thinking about Brownian motion as just a discrete random walk with the mesh size approaching 0 can get you quite far.