Acidic character and anion stability across periods and groups of the periodic table I understand that to compare relative acidity one must consider the stability of the conjugate bases.
Across a period the electronegativity of an element increases. And that is for example why $\ce{HF}$ is more acidic than $\ce{H2O}$, as the negative charge of the fluoride anion is drawn closer to the positively charged nucleus and thus more stabilized than that of an oxide anion.
Down the group the atom radius increases. And that is for example why $\ce{HI}$ is more acidic than $\ce{HF}$, as the negative charge of the iodide anion is distributed over a greater volume and thus more stabilized than that of a fluoride anion.
To me, these two trends are contradictory.
How can the stability of an anion increase as the negative charge gets closer to the nucleus(observed trend across periods) but at the same time decreases as the negative charge gets further away from the nucleus(observed trend down groups)?
Is there a concept I have misunderstood?
There are two factors determining acidity of a compound, the electronegativity factor and the size factor. Down the period, the size factor dominates, and the reason $\ce{HI}$ is more acidic than $\ce{HF}$ is because of this factor, as in the former molecule, the orbitals of iodide participating in bonding are larger and more diffuse, hence bond is weaker.
Across the period, however, size does not vary as significantly as down the group. Hence, we compare using the electronegativity factor. If electronegativity difference is higher, there is larger ionic character, and hence it is easier to deprotonate.
Also, basicity of an ion is proportional to its charge density. So you misunderstood the first concept. You should compare fluoride with hydroxide ion, and not oxide.