The modes are found on the Thermomix screen by swiping left. Scroll down for more modes.
My most used modes are the Scales, Chopping, Pre-Clean and Kneading function.
I suggest you occasionally check for updates as new and exciting things will be added regularly. For example Peeler was added only this year as was the fast wash.
I rather love the egg boiler mode, it is such a relaxing way to boil your eggs and you don't have to hunt in the cupboard for an egg timer. Despite its instructions I find it works just fine with eggs which are not kept in the fridge, and with large size eggs.
Possibly the Thicken mode is the cleverest, particularly useful for those of you who like to use your own recipes. Read my explanation on my Thicken mode page here
You can re organise the modes to keep your most used at the top of the screen