This letter-change at the beginning of words is called "mutation" (treiglad in cymraeg) and it's a wide grammatical field, and yes, there are rules of when and how to mutate. What you noticed in mynd and menyw are both examples of the "soft mutation", which is by far the most common mutation, and I can only tell you which letters change to which, but I won't try to list any rules here, because a) that would be too long and b) I don't even know all the rules myself.
The most important thing is to notice that these things happen, and as you become more exposed you will develop a "feeling" of when and how to mutate, and if you get it wrong, don't worry, you will still be understood.
Now here is the list of letter changes in the soft mutation:
p -> b, m -> f, b -> f, c -> g, t -> d, d -> dd, ll -> l, rh -> r, and as you noticed, g just drops completely.
For more information I can refer you to this web-page: https://en.wikibooks.org/wiki/Welsh/Mutations, or the book "Modern Welsh Dictionary" by @garethrking which gives a short overview.
With what I wrote above you can now see that ddweud is just the soft-mutated form of dweud, so they are in essence the same word.
Now dweud wrth in english is "to tell", which is the action of saying something to a specific person, but like you said, italian only knows dire for both actions, so this is also one of the things that you will pick up naturally as you go along, but again, it's not the end of the world if you "get it wrong" from time to time.
And: Llongyfarchiadau mawr iawn am gwpla lefel 1! (Congratulations on finishing level 1!)