Hi and welcome to the forum. As the others have said, things work a bit differently in Thailand when it comes to accommodation. Whilst in the UK or Oz a budget traveller essentially has the choice between budget hostels with shared dorm rooms and more expensive private rooms in hotels, in Thailand there is more of a smooth grading between the two. Since you can get a cheap private room in a guesthouse for a few hundred baht, it doesn't make sense to have the annoyance of sharing a room with ten other people all coming back late, making noise, putting on the light, getting up early etc and ruining your sleep just for saving a pitiful amount of baht.
The one thing dorm rooms ARE good for is meeting other travellers - you can't help but not chat to your roommates, and if you are keen to meet others then I would suggest finding a newish, clean hostel offering dorms is a good idea (
Soi 1 Guesthouse in Bangkok seems to come well recommended, if you're spending any time there either side of Phuket - a solo traveller mate of mine stayed there the other week and judging by his facebook photos, he had a great time!). But since it's you and a mate, why not go for the best of both worlds: book a twin room to save money, but choose a guesthouse with a lively communal area (check hostel reviews) that makes it easy to meet others?