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The Laugarvegur is generally walked as a four-day trek from Landmannalaugar to Þórsmörk, although the speed record is a mere 4 hours 20 minutes for the whole thing! In recent years it's become common to add a couple more days by walking the Skógar Trail from Þórsmörk to Skógar. Less well-known is a recently-marked three-day trail allowing you to walk from Rjúpnavellir to Landmannalaugar, so that the whole thing can now be covered as a nine-day trek.
In the summer, generally from mid-June to early September, Reykjavík Excursions buses operate a daily service to Landmannalaugar (with the option of getting off early at Rjúpnavellir), as well as daily buses from Þórsmörk back to Reykjavík. The Trex bus company also does this, slightly cheaper, but only through July and August. If you continue to Skógar, then Sterna buses run daily along the south coast between Reykjavík and Höfn, both ways. This summer, the road to Landmannalaugar didn't open until 22nd June, and the highest parts of the trail were completely buried in snow. This year, for the first time, there was a bus running on alternate days that linked the Laugarvegur huts at Alftavatn, Hvanngil and Emstrur (and also the youth hostel at Fljótsdalur) in a circuit from Reykjavík, but there's no guarantee the service will operate next summer. I gather it wasn't particularly well-used.
If you're capable of using a map and compass, following waymark posts and a reasonably well-trodden trail, then you can walk the Laugarvegur and Skógar Trail without a guide, but if you want someone to transfer all your stuff from hut to hut, then it's best to sign up with a company that's offering a guided trek with this option. If you want to stay in the huts during July and August, then book them NOW, because they do get filled to capacity in the peak season. If you want to camp, then there's no need to book. Bear in mind that if you book and are tied to set dates, then you'll have to take the weather as it comes, and it can get really nasty. If you don't book, but camp, then you can choose to walk in a spell of good weather if you give yourself enough days to be flexible while you're in Iceland. Honestly... walking ANY Icelandic trail in lousy weather is fairly pointless... it's all about views and stunning scenery.
When you're through with the trek and want to hire a car, you'll find it's pretty expensive. The usual trick is to find another couple to share costs with. If you get the cheapest possible deal, the car won't be allowed on any dirt roads, which will basically limit you to the ring road around Iceland, and a few roads branching off it. If you damage the car on a dirt road, or crossing a river, you have to pay for it. You'll be able to see some nice waterfalls and do some short strolls, but lots of exciting walks will be out of bounds to you while you're motoring.
I met THREE honeymoon couples on the Laugarvegur this summer... so don't imagine you'll be the only ones!
