Which set is bigger, the natural set or the words\string set?
Let's say a word is any sequence of letters.
The posible number of words size 1 is $26^1$, and for 2 is $26^2$, etc. From that, we can say that the number of possible words is $\sum_{i=1}^{|\mathbb{N}|} 26^i $.
So, we can use all the naturals just to enumarate words that only use a's;like "a", "aa", "aaa",etc; since thera are $|\mathbb{N}|$ of them, and still need more for the ones with b, c , d...
But, if we asign each letter a number, like a = 1, b = 2... z=26. We can see a word like a Natural number in base 27, for example "abc" is $a(27^0)+b(27^1)+c(27^2)=1(27^0)+2(27^1)+3(27^2)=2242$
That way, we have a can pair all words with an unique Natural, and still be missing numbers that have zero on it, like $20_{27}$.
So... what's the answer?
Infinite sets are weird, and we have to be very careful and precise when asking questions about them - even when just talking about "simple" notions like size.
When measuring the "size" of arbitrary sets, we compare them in terms of bijections: two sets $A$ and $B$ have the same size (cardinality) if there is a bijection between them. If there is an injection from $A$ to $B$ but there is no injection from $B$ to $A$, we say $A$ is strictly smaller than $B$.
This isn't the only way one could try to talk about the size, in an informal sense, of an infinite set, but it turns out to be by far the most generally useful. If you want to compare sets in a different way, you can definitely do so, but then the onus is on you to clearly define what you're doing.
Now what you've observed is that you have a pair of sets with injections going both ways. It turns out that we can turn this into a bijection - the two sets in question in fact have the same size! This may seem counterintuitive; I think it's a good idea to start small, and think about why the set of natural numbers and the set of even natural numbers have the same size.
(Once you've understood this, you'll be able to really appreciate the existence of uncountable sets.)