初めまして　or　始めまして？
When I write Japanese on my computer I get bot options to chose from. When I tried to resolve the problem by checking which means what, I was surprised to find out that both of them mean "How do you do?". Help! How should I write はじめまして? By using 初 or 始?

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There is clearly difference.

初め = first
始め = beginning, start, launch

Difference between those two is "FIRST" and "BEGINNING or START". Hence, "how do you do?" goes to the former but not the latter one. Now you know what comes next?

When you want to greet someone you've FIRST met, you will say
初めまして = nice to me you, how do you do?

When you will START or BEGIN a new business, you will say
あたらしい　しごとを　始めます。

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I use 初めまして if they have never met me before. Actually both are correct according to online dictionaries and my dictionary, but I think  始めまして is less common nowadays.  Also I often write it in ひらがな.

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初めまして → Nice to meet you. 
始めまして → I'm starting.

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始め　and 初め are just two different ways to write the same word, which of these 2 characters you have to use, depends on the context. In case of the greeting "hajimemashite" it's the second option - 初めまして. Hope this helps.

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初めまして → Nice to meet you.
始めまして → I'm starting.