Hi,
You can always use free blogging host or forum. But free hosts come with a price. Search engines take self hosted sites more seriously. besides, more importantly, visitors take your site more seriously if they see that you are not on any free host.
take a test yourself. Which one will you like more as a visitor?
xoogle.com
or
xoogle.bloxspot.com
so, i recommend, you go with paid host and domain to host your forum/blog.
forum softwares can also be found free or paid. Paid is vBulletin which is great (but costs alot) and a free forum software is SMF (great considering it costs nothing and superbly flexible). You can use many "mods" to make your main site (homepage) display info like a blog. You can use portals in forums to do that.
SMF site example: http://bangladeshtalks.com/index.php
vBulletin site example: http://www.cricket-highlights.com/
so my recommendation is:
1. use a paid host to host your own doamin. godaddy.com is what i use; ($56.88/yr, 10GB space, 300GB Transfer)
2. Use SMF 2RC2.0 as your forum software which is free to use.
3. Use a portal in your forum if you want its home page to be blog like.
4. or if you decide to host a blog use wordpress blogging software. it is great.
i hope this helps you.