[u]Here[/u] is the blink without delay example.
Note that you can use multiple timing variables. In your situation, you need one "delay" (actually a "time reference" rather than a delay) for the chase speed, and another time reference for the speed-change rate (i.e . 10 per second).
It would be a good idea to review the entire [u]Language Reference[/u] to get an idea of what you can do. You don't have to memorize the whole thing. But, try to understand if-statements & loops. If-statements (and other conditional logic) are the way that computers "make decisions". Along with loops (doing things over& over, perhaps changing variable-values each time through the loop) these are the two most important things in programming... They are the things that make programming useful.
Let me say that I don't know how to write or read code, I learn best by taking something and changing values to see what I changed, then things me more sense.
That's not a bad approach... When you are modifying code, it's a good idea to change a little at a time. When something goes-wrong, it's not always easy to figure-out what's wrong and it you change one or two things at at time, at least you know where the problem is.
The BEST way to learn programming is to take a class. The 2nd best way would be to get a book. But, I don't know if you can find a class or book that teaches beginning programming concepts as well as Arduino programming. So, unless you want to work through a general-purpose C/C++ programming book (unrelated to the Arduino), you might just have to struggle-through.
The same goes for writing code from scratch. Professional programmers don't write the whole program before trying it out... They write and tests one thing at a time. Beginners should try to write and test one or two lines at a time, except where you need to add more to make the program comple & run.
Just FIY - Programming is NOT easy, and it's probably different than anything you've done before. People who are good at math & logic tend to make good programmers. Just as an indication of how difficult programming can be, professional programmers make mistakes (create bugs) every day! If doctors or bridge designers made as many mistakes they would be fired! 
(Most of these bugs are fixed before anyone sees them.)
I've been programmed (part-time for work and as a hobby for many years in several different programming languages... I'm still not an expert!