There are numerous regular sound correspondences between Hungarian and the other Uralic languages. For example, Hungarian á corresponds to Khantyo in certain positions, and Hungarian h corresponds to Khanty x, while Hungarian final z corresponds to Khanty final t. For example, Hungarian ház "house" vs. Khanty xot "house", and Hungarian száz "hundred" vs. Khanty sot "hundred".
The distance between the Ugric and Finnic languages is greater, but the correspondences are also regular. The relationship is most obvious when comparing all the Ugric languages with all the Finnic languages, for then individual idiosyncrasies are averaged out, but here we will just compare Hungarian with Finnish and Estonian. One important difference between Hungarian and the Finnic languages is that Hungarian underwent the most salient features of Grimm's Law while the Finnic tongues did not.
Hungarian [f] corresponds to Finnish and Estonian [p] (just like English [f] in father corresponds to Latin [p] in pater):
Hungarian
Finnish
Estonian
meaning
fa
puu
puu
tree
fél
pelätä
pelgama
to fear
fészek
pesä
pesa
nest
Hungarian [k] corresponds to Finnish and Estonian [k] before front vowels:
Hungarian
Finnish
Estonian
meaning
könny
kyynel
küünistama
tear
kéz
käsi
käsi
hand, arm
kő
kivi
kivi
stone
Hungarian [h] corresponds to Finnish and Estonian [k] before back vowels (just like English [h] in hound corresponds to Latin [k] in canis)
Hungarian
Finnish
Estonian
meaning
hal
kala
kala
fish
ház
kota
koda
house (Hung.), hut (Finn. and Est.)
húgy
kusi
kusi
urine
Hungarian [t] corresponds to Finnish and Estonian [t] at the beginning of a word:
Hungarian
Finnish
Estonian
meaning
tél
talvi
talv
winter
tud
tuntea
tundma
to know
tavasz
touko
spring
In the middle of words (note that due to the loss of the word final vocals in the Old-Hungarian language these are now at the end of the words) Hungarian [z] corresponds to Finnish [t] (which can alterate with [s]) and Estonian [d] or [t] (which is also able to alterate with [s]):
Hungarian
Finnish
Estonian
meaning
ház
kota
koda
house (Hung.), hut (Finn.)
kéz
käsi : käte-
käsi : käte-
hand, arm
fazék
pata
pada
pot
This is just a sample. Even in the small number of words above, other regular sound correspondences can be seen, such as Hungarian [l] corresponding to Finnish and Estonian [l].