"How do I count the rows accurately in my active work? Do I include the cast-on row and the stitches on the needle(s) or not?"
A 2-part question gets a THREE-part answer...
1. COUNT THE CAST-ON?- If you cast on with a loop method (such as backwards loop or forwards loop) the first row you knit is the first row of the work--the loop cast-on is not usually counted as a row. (This is a convention because, if the cast-on were to be counted as a row, pattern writers would have to write a different set of instructions for the first repeat of a texture pattern counted in rows.)
- If you cast on with the long tail method, the first row is knitted at the time of the cast on. Therefore, the first row you knit (or purl) after the cast-on is actually the second row of the work. (More detailed explanation at the long tail post.)
- With a cable method (also called "knitting on" or "chain cast on") it's knitter's choice -- this kind of cast-on is heavier than loop cast-on, but not quite as doubled as long-tail.
click picture

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2. COUNT THE STITCHES ON THE NEEDLE?
Now, despite all this hyper technical stuff, reality is that one row more or less might make a horrible mess of a TEXTURE pattern in a fabric, BUT -- one row here or there is unlikely to make a difference in the FIT of any knitted garment. Excluding the cast-on row from your row count makes no noticeable difference in the finished garment. What WILL make a difference is CONSISTENCY in counting rows between the different parts of your project.
Example: suppose the front of your new sweater is knit 76 rows to the underarm, NOT counting either row of a long-tail cast on as the first row, and NOT counting the stitches on the needle (in other words, not counting the red, green or blue rows of the illustration.) Now suppose the back is knit 76 rows to the underarm. This time, you DO count the rows you didn't earlier (in other words, this time, you do count the red, green and blue rows of the illustration.) The front piece would wind up 3 rows longer than the back. Awkward at seaming time.
Bottom line: There is a convention for which rows to count as part of the row count. However, to achieve a good fit, CONSISTENCY in counting is the most important thing--much more important than whether you choose to follow convention.
3. ADDENDUM:
Even though it's not part of Angie's question, the title raise the different question of keeping track of rows as you knit them.There are lots of ways to keep track. Clickers of various kinds are popular.
Whatever way you keep track, however, the one certainty is that you will lose track. The phone will ring, your kid will crash into something, your city council rep will ring the doorbell. This is why "reading" the fabric, as Angie wants to do, is your best insurance policy.
--TK
(You have been reading TECHknitting on: How to count rows)