Is there a place on the internet that gather stories of rpg game sessions? I've read some on game-specific websites but is there a common place to read other's tales?
For improving your searches in the future, these have a technical name that is frequently used online: Actual Play reports (a.k.a. AP reports or just Actual Play or AP). When searching for game reports online, it never hurts to run another search including the search term "actual play".
There is a sub-forum at story-games.com called The Game Report that is specifically for posting AP reports and for collecting links to AP reports on other sites. These tend to be more story games reports, like the site says on the tin.
RPG.net has a dedicated AP forum called Roleplaying Actual Play that tends more strongly to traditional roleplaying games. The titles played are pretty wide-ranging, so this is a good place to look for reports on games that are solidly-mainstream (4e, Pathfinder) to fringe-of-mainstream (Mythic, Apocalypse World, Nobilis).
The RPG Site has a mixed sub-forum devoted to actual play reports, dissection thereof, and non-theoretical game design discussions called Design, Development, and Gameplay. It's a bit more hit-and-miss because of the mixed purpose of the forum, but browsing the thread titles will usually tell you which are gameplay and which aren't, when they're not explicitly tagged '[Actual Play]' in the title.
I find that reading an Actual Play report is always more informative about a game than any reviews I can find, and these are my go-to places for that, in that order.
For Old School D&D, Dragonsfoot has a thriving "Campaign Journals" section:
Obsidian Portal is a gaming-specific wiki, and part of the site includes space for an Adventure Log. Here's an example one from the current Featured Game.
RPG Geek has actual play forums for all games in the DB. Not all have been utilized, but all are set up for it.
Pick the game you want, and look for content. Or add your AP reports.
Pen & Paper Games is a place for all things RPG. There is a particular blog on here about a Barbarian named "Og" (and his party ofcourse). He is hillarious and it's a must read. It's been awhile since I first read it but I laughed to tears when reading some of them.
Here's the link to the DM's posts. There are other blogs on here aswell, if you want to browse around. Hope this helps. Pen & Paper Games
Just today I had found this video. I don't know much the others, but one of them is Mark Rein·Hagen, Vampire the Masquerade and Ars Magica creator, and posibly my favourite game designer. I have faith in that he roleplay very well, although I have not seen the entire video.
Hope that helps giving a fine example about how roleplay works.