Onegai shimasu and Onegai Itashimasu

When meeting and expressing "nice to meet you", can it be said as "yoroshiko onegai itashimasu" instead of saying "yoroshiko onegai shimasu". Will it be extra polite to say this in case of a formal situation ?

yes. Itashimasu is more polite than shimasu :)

Thanks, i understand that it is alright to use "itashimasu" even while introducing yourself.

yes. Nothing wrong with using it in any context.

What is the difference between ga hanasemasu and o hanashimasu. Nihongo ga or o...

Hanasemasu - I can speak Hanashimasu - I speak or I will speak. Hanasu is the main verb. It's present form is Hanashi, while it's stem form ( often used as a potential form ) is Hanase. Ga makes Nihongo the subject of the sentence, while o (wo - を) makes it the object of the sentence.

コッドーサン　has already explained very well. Just to add, "potential form of verb" is actually a form of verb in which the ability or potential to do something is expressed. So when you are expressing about the potential to do something, you are actually talking about something which you can do, which makes it the subject of the sentence. For subject you need to add 'ga', the topic marker particle.