Once you cross over at the Eilat/Taba border, you’ll find numerous taxis. Opt for a shared taxi, it’s definitely the easiest and quickest method of getting to Dahab.
Once in Dahab there are a few Agents that can take you to Mt Sinai & St Katherine's Monastery. Some people climb the Mount at night to watch the sunrise while others do it in reverse. Either way is spectacular.
St Katherine’s Monastery is sometimes closed due to religious holidays so please check opening hours before making the trip. The Monastery is closed Friday and Sunday, so if you’re planning on a sunrise trek to Mount Sinai ensure you work around the Monastery morning opening times. For example, climb Friday night so you're at the Monastery Saturday morning ...
For the more adventurous traveller there is a Bedouin Bus service which runs between Dahab and St Katherine’s linking the communities. This is a sponsored project and offers a very cheap way to get between Dahab and St Katherine’s.
http://www.bedouinbus.com/
I personally recommend FEEL LIFTED. This is a wonderful family run establishment located on El Fanar Street in Dahab. Maggi May and her Bedouin husband Mohammed offer one-to-one service and tailor made trips either for single days or overnight.