You can do this currently by going to Administration > Site Settings > Permissions > Roles that can upload and browse but only in a user specific location, and check the box for "Authenticated Users"

Then users can upload images in the forums using the image toolbar item in the editor and can upload files and link to them from the link dialog in the editor which allows browsing/uploading files.

Best,

Joe

How do I modify the allowed file types? 

Look at the settings in Web.config for allowed file extensions, you can copy that to user.config and modify it.

Are there any plans to add a browse and attach file directly to the forum post page?

That is what this allows, users can upload a file and create a link to the file, it isn't an attachment in the same sense as email has attachments because this is not email it is a forum and a forum is just a web page in a web page you can link to files. There is no different implementation planned. If by attachment you mean you want to have an icon next to the link like a paper clip that could already be done by some clever css based on file extensions, no feature needs to be implemented to do that.

Best,

Joe