Well, I played my first ranked game today, and I won. (Carried my team with super-powered mundotank, but that's beside the point)

My ranked rating is showing 1 win, but my rating is "unranked".

Does this change eventually, or do I need to win/lose more games?

When you start playing ranked you always start at the ELO of 1.200. This is the ''starting point'' for all players. The ELO hell is normaly mentioned for those under 1.2k ELO.

Anyhow.

When you start playing ranked you will have to play some matches before you ranked stats will appear in the upper right corner after the match and/or your summoner profile.
You still recive the ELO of a win or a loss each match but you wont see the +/- ELO you get till you've played those few matches.
I think the number or matches you have to play is 10. I'm not 100% sure on this fact though. But once you have played those 10 you will start to see your ELO.
Notice that if you've wond all those 10 matches wou will have like 1.4k ELO.

Soo..You aint really unraked..It's just that you have to play those matches intill you see your ELO.
You still gain and loose ELO though so play carefully

Cheers
Sawy-Janna player

i bet riot is putting you in the top10 as we speak

Basicly you are not gonna get ranked by the first few games.

You need to:
1, have played more than 10 games in ranked (not 100% sure about the exact number)
2, have rating of more than 1200 (less is not shown)

Its after 3 games tough.