Here's some stuff you'll see a lot right when you start out, so I thought I'd explain them here. Pronunciation of the kanji is in katakana.

大 丈夫（ダイジョーブ）This is 丈夫 "durable" and 大 "big", so we end up with "Hey, I saw you fall! Are you alright?" "Don't worry, I'm VERY DURABLE!" = I'm fine!. Thus "Hey, are you really durable?" = "Are you alright? Can you handle whatever's happening to you?".

お 邪魔 し ます（ジャマシ） The お means "you(r)", 邪魔 means "hinder", します means "(I) do/am doing". So together "I'm doing something that hinders you" = I'm being a nuisance/bother (to your work, to your peace at home, etc). 邪魔 する な！ "don't hinder!" = "stop bothering (us)!" ex. don't break into our conversation!

失礼 し ます（シツレー） The 失 means "lost, miss(ing)", 礼 "bow (= greeting), thank(s), manners". So "(I'm) lacking greetings/manners" = "I'm being rude".

お 早う ござい ます（ハヨ） This is お "you(r)", 早い "is early, fast", ござる "exists". Aka "You're up early! It's early to be seeing your face today!" etc. even if it's not actually early.

ありがとう ござい ます
お めでとう ござい ます
In archaic grammar, い-verbs all change to おう when directly in front of ござる but in the modern standard language this is just left in a few set phrases like "congratulations (おめでとう)", "thanks (ありがとう)". You WILL end up seeing the original forms おめでたい and ありがたい if the words are in (implicitly or not) front of something like です instead of ござる. I don't know for sure but the difference seems to be ex. ありがとう "I'm thankful to YOU", ありがたい "I'm thankful to (someone/something other than you, ex. the painter of this painting)".

今日 は（コンニチ）
今晩 は（コンバン）
Both of these are この "this" (technically こ this もの object) + a time of day. So today is "this day", tonight "this night". Then we're just not saying the full sentence "This is a beautiful night to meet you on!" or whatnot which just happened to turn into a greeting at some point.