So is this right?
 

Im going to speak Japanese in Romaji. Is it right so far?

It is pretty short, but I want to know if someone in Japan would say that regularly.

No. "Hajimemash'te" is not a word. Also, learn to use kana. Most natives here will ignore your writing if it is in romaji.
InputKing online IME - type your language on any computer
There, now you have no excuse not to use kana.

Oh ok. Sorry

Oh. Thanks!

ignoring the romaji for a moment, you have two different greetings there, which are used at two different times. I don't think you ever use the two together. Separately, both are quite common words in Japanese.

こんにちは　konnichiha (the 'ha' is pronounced 'wa' in this case) means 'good day', and generally implies that you know the person your greeting to some extent.

はじめまして hajimemashite is what you say to a someone you don't know when you first meet them. Typically it's then followed by a jikoshoukai じこしょうかい　or self-introduction. Or someone might make the introduction on your behalf.

But go study up your kana before you go too much further. There's lots of websites that can help you if you google.

On a tangential note, I don't know how many people actually think about this, but it's painfully obvious when it gets pointed out: はじめまして is the polite continuative (て) form of はじめる, which is the transitive verb "to begin [something]" (as opposed to the intransitive はじまる, which means "[something] begins").

Furthermore, こんにちは is an alternate pronunciation of 今日は, which could be translated literally as "Today/This day is...?"

Right, はじめまして literally means "We begin" more or less. And こんにちは is part of a longer archaic sentence where the end has dropped off over the years. Which I did know in it's entirety, but damned if I can find it again now. SM knows probably.

I believe it was こんにちはご機嫌いかがですか？
But if I am wrong I'm sure I will be corrected :)

And おはよう comes from お早い ("early"), ありがとう from 有り難い ("difficult to exist"), すみません is 済みません ("it will not end"), etc.

Isn't etymology fun?!

I wonder if there's an online Japanese etymological dictionary like this amazing English one: Online Etymology Dictionary