I created a macro for informing parties when I was new to a dungeon. Very simple button on my bars that basically just said "/p Hello. First time here. I apologise for any mistakes. Let me know if there's something I should watch out for." Then I just hit that button upon entering the dungeon rather than having to sort out typing in chat, eat food, get pets out, etc.

As for dungeon runs - sounds like you're doing alright - follow tank, kill what they attack, get out of the way of yellow areas, avoid flashing canisters. Dungeons do get easier (and more fun) as you progress and level but don't feel that you need to keep quiet. FFXIV has one of the best (and nicest) communities around so the majority of people wont ridicule you for asking questions or asking for help.