In summer, from about late June, there is a daily bus service from Reykjavik which lasts several hours but takes you through some some of the most stunning landscapes you are likely to encounter in a bus. You start off in the coastal lowlands but quickly move to dirt roads and eventuly barely discernable tracks across the pitted lava and ash plains in the shadow of Hekla, one of Icelands more active volcanoes. In high season July and August, bus services are also available to/from Akureyri, Myvatn and Skaftafell.
You can also attempt to reach Landmannalaugar by sturdy car, although a 4x4 is strongly advised check your hire car insurance - it may specifically exclude the "highlands" of the interior. There are three roads to Landmannalaugar. The most easily passable road is from the north, via the paved road to Hrauneyjar, and passes through desert scenery typical of the deep interior. The road from the west has several fords which may be passable in an ordinary car depending upon the water level. The road from the east is the roughest and has the most fords. There is a deep ford immediately before the Landmannalaugar site which is not normally crossable in an ordinary car, but you can park before it and arrive on foot.
Landmannalaugar is only accessible in summer. The dates that the roads open vary from year to year. Typically the road from the north opens earliest, sometimes as early as late May, more usually by mid-June. The western approach usually opens in June, and the eastern approach usually by the end of June. The roads usually remain open through September, and road access may finally be lost any time from late September to early November.