It takes me at least 3-4 hours (cumulative) to learn 5-10 psukim (sentences) in trope. This is just one Aliyah! And unlike many of you, I know the notes, the grammar, and understand most of the words. Yet it takes so long. I remember as a child that my Orthodox Grandfather z"l prepared and was able to learn & chant (lein) an ENTIRE Parasha in practically no time. What am I missing?
- Understanding the trop will help immensely (i.e. which is a "mesharet" - helper vs which is a "Mafsik")
- Many parshiyos have very similar words with slightly different trop, keeping track of those and trying to remember the different tunes will help.
- Trying to understand the pasuk being leined while leining it can help (this can be pretty hard for beginners)
- Using a good token Korim can make a huge difference (for one, the צורת הדף will be very similar to the one in the actual torah , for many torah's)
For all the above I would greatly suggest the Tikkun Kotrim called "Simanim" by Rabbi Riachi. It has a terrific intro explaining some basics of dikduk and trop, and he came up with a good way to point to many items to remember withing the text of the pesukim.
You can purchase it here (if you are sefardi, make sure to get the nusach עדות המזרח version)