With this number of users I would strongly recommend taking a VPS rather than shared hosting - it sounds like this would be within your budget too... assuming that the figures quoted are monthly?
A VPS will provide the level of resources that you need to run a VPS of this size, but particularly relevant here is that a VPS will be most suitable for supporting your expected growth. Check with your intended provider, but technically speaking your VPS should be able to grow as required without any downtime or any other potential inconvenience (e.g. no migrations causing a changing in hosting environment, and the potential problems which can occur as a result of that such as different versions/settings in PHP etc.).
Also worth considering in your case is the upgrade path to a dedicated server; you should be able to upgrade from your VPS to a dedicated server. Check that this is also possible with your provider, and certainly make sure that you have some plans as to how this might happen if/when the time comes.
I noticed that informe.com are using phpBB - that's very good for you, as it means that you can easily backup your database (via the phpBB admin panel) and export your forum onto a clean install of phpBB on another hosting service.
I've already encountered various examples of what you mean regarding the speed of their servers!
If you need any more assistance with this then please feel free to PM me. I'd be happy to talk through it all in more detail, but I have a bad habbit of creating long posts... no point me explaining it all if you're already sorted out