I don't know if it can be useful in your case, but there is a general theorem that states that, given a cardinal $\displaystyle \kappa \geq \omega$ such that $\displaystyle 2^{< \kappa} = \kappa$, then there exists an almost disjoint family $\displaystyle F \subset P(\kappa)$ with $\displaystyle |F| = 2^\kappa$.
This theorem, with $\displaystyle \kappa = \omega$ answers your question.
Come on, do not take a sledgehammer to crack a nut. There is a simple example:
Let the $\displaystyle P_i$ be such infinite set that $\displaystyle p_i^k \in P_i$, where $\displaystyle p_i$ is the i-th prime number and $\displaystyle k \in \mathbb{N}$.
Then infinite family $\displaystyle \{ P_1, P_2, P_3, ... \} $ is an example satisfying the requirements. Because for every distinct $\displaystyle P_i \cap P_j = \{1\}$ and $\displaystyle P_i \subseteq \mathcal{P}(\mathbb{N})$ and there are infinitely many primes.
I read a problem like this somewhere.
since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.
I actually did a similar problem to this (as you described). The question was "Does there exists an uncountable family of subsets of a countable set such that the interesction of any two is finite?" The answer is yes. I gave two examples. I actually did it on the forum if you are interested. But! The OP said that the sets need to be subsets of $\displaystyle \mathbb{N}$ and as your sets are not this doesn't work. I refuse to actually answer the question upon the grounds that the OP has NEVER shown any effort.