Slims do have removable hard drives, they just slide out the back. There's like one screw you remove and pull it out.
I'm guessing you have a low GB HD, I came close to filling my 120GB HD on my model 1 XB360 then found a Slim at a Garage sale with a 320GB HD and transferred everything over and now use that.
If you have a Model 1 you can upgrade to a 250GB HD but you'll have to check out used game stores to find one now a days.
If its a slim, pretty much the same, most used game stores sell HD's just find one that's at least 250GB - 320GB. I used 250-320GB HD usually sells for $30-$50. Due note you're going to need a transfer cable to move your data to the new HD. especially if its an older version.
But if you have a 250-320GB HD and have it filled it, odds are you have downloaded a ton of Full Games, DLC and movies that take up a ton of space, then you'll have to manage your storage and delete what you don't use anymore. You can re-download anything that you buy on your account. But I don't recommend deleting any promo DLC you picked up. a lot of those are removed and can't be re-downloaded later. (some can and others you'll have to pay for it.)
You may be able to use the Cloud to store extra Games, DLC & movies, but I personally have no experience using the Cloud.
A flashdrive works for game saves, but your not gonna be able to transfer games, DLC and movies to it. Most game saves don't eat up a ton on HD space unless you have a ton of games like i do. lol
And some games will not let you transfer to any memory device.
I knew a guy that had a 20GB HD for years and he was always managing his saves and DLC. Deleting anything he didn't play anymore.