You can name your company one thing but "trade" as any number of names that you would like. So the official name of your company would have "LLC" at the end of it, but if you want your company to be known by just its name without the LLC use whatever name you want.
When you have the budget it would be a good idea to get a trademark secured for your company name the way you want it.
You need to use the company name with suffix (LLC, L.L.C., or Limited Liability Company) on all official documents (contracts). You can use a trademark with any name without the LLC - so on your website it can just say Company Name without the suffix - but you should make sure you're not infringing on someone else's trademark.
If you don't use the LLC in the name, the filing may get rejected by the filing office. Just use the LLC in the name, and then if you want to have a "trade name" without the LLC, just use whatever you want and file a "certificate of assumed name" or the like with the same filing office.