> So there are infinite many infinity but let's talk about the two most common one, countable and continuous.

I think you mean "uncountable", not "continuous". Or possibly "the continuum", which is sometimes used as a fancy word for the real numbers.

> Countable is any thing one can count, think of the number 0,1,2,3,4...78810836689017,.... This will go on forever thus infinite, but in a infinite amount of time one could count all of the numbers (this is not possible for human because we have finite time).

I don't think it's helpful to talk about what you could do in an "infinite amount of time", because like you said that's not possible, and it's not really obvious what we might be able to do given infinite time.

A more concrete way to talk about this is to say that, if we have a countable set of objects, we can come up with a way of listing them that will eventually reach any given object. For example, if we list the positive integers like 1,2,3,4,..., then you can pick any positive integer you like and it will show up in our list eventually.

> Continuous think of decimal pick two decimal call the larger one B and the sampler one A, then pick a decimal C; where C is in-between A and B, then repeat (C will now be B and one will pick another C) This will go one forever and you can always find another decimals that we didn't account for. One can't "count" all of the decimals because you can always pick another decimals between A and B.

But this just shows that your particular approach to counting them didn't work. Maybe there is a way of counting them that doesn't keep zeroing in on a smaller and smaller interval. In fact, for the set of all numbers with finite decimal expansions, there is a way.