A few people have emailed me asking about transport arrangements and hotels in Tamanrasset. I booked my flights from the UK with British airways who fly London Gatwick to Algiers a few times a week.
All my arrangements in Algeria and in Tamanrasset were made by Timgad Voyages, a specialist tour company based in Batna, Algeria. They have been operating since 1981 so are experienced with a good reputation. If you don't speak French or Arabic you will need an English speaking guide and their best one is Redha who was my guide on both my trips to Algeria. His English is perfect and he is an excellent guide. We flew from Algiers to Tamanrasset with Air Algeria. We also had 2 expert desert guides for the trip into the desert who provided all the food and equipment as well as 4 wheel drive vehicle. Unless you know the area well it is essential you get a desert guide who knows their way around - it is very easy to get lost in the desert and this is proper desert - no signposts or cafes, and the only people you meet are Nomads who speak the local language of Tamashek.
Timgad Voyages website is http://www.timgad-voyages.com/EN/ATV/Presentation.php
This is also a useful site for information.http://www.world66.com/africa/algeria/tamanrasset
Tamanrasset and Djanet are the two algerian pearls, starting point for discovering the deepest saharian locations. If on the map both cities seem close to each other and located on ancient caravan roads, they are difficult to bind. From Tamanrasset or Djanet, no busses are linking the two places. If you want to see the two places in one trip, the best method is to catch a plane (single flight for ~40€). By the tracks you need to combine a trip with a travel agency, but you will have to pay for the back trip even if you are not in the cars - the time of transfer at maximum speed is of two days...
Best option is probably to visit the places between the two cities and take a week between the two places : ask a local agency for organising all that.
This is the low cost solution to reach the southern city of Tamanrasset, you can cross all the northern Sahara (Algiers, Ghardaïa, In Salah, Tamanrasset) by bus. Total cost is around 30-35€. However the trip has some limitations : (1) time! two days are necessary to cover ~2600km, and add at least another for relaxing after this long trip! (2) confort! small size for seats in buses, so if you are - like I - ~1m85 and 90K, I would suggest something else. Then, the road from In Salah is of bad quality and you may suffer during some uncomfortable hours, don't do that if you haven't already experienced a 12h trip in e.g. Nepal! (3) security! Even if the police stands everywhere, plan your trip, halts, and overnight places in advance. Avoid improvisation during this long-and-short trip.