I have just bought a smartflash kit and now need a bag to store and transport all the bits I already have and the additional bits I have bought.
Any suggestions of a bag that will protect the kit but is not very expensive?
You have a ton of options, the two obvious ones in my view are:
A large trolley type to take everything which has the disadvantage of being very heavy and bulky to store or move through small areas but is the easiest way when you're moving it over a flat surface as on a properly designed one the weight gets nicely balanced so you don't have to exert much effort to pull it along. A good example of this would be the newer Bowens trolley bags (either the latest version (also sold under the Limelite brand) or the one with a separate trolley, not the version with the wheels you have to screw in they're cheap but don't work half as well).
Second hand you'd spend £50-100 depending on the version, Calumet also has (had?) some reasonable options, Interfit do a reasonable trolley case which is also sold by a few other companies like Hansel etc but most people I know don't like this option as a full mini-studio gets very heavy.
Two bags, one for your heads with small accessories and one for your stands/brollies. The advantage is it's easier to store and move around as there shouldn't be much wasted space and if you've got a few stands the weight will probably be split evenly between the two bags (you can of course mix and match wheeled versions, Elinchrom/Bron etc do a padded roller case for 2 heads and there's plenty of wheeled stand case options). The cheapest option for the stands as mentioned is a hockey bag (cricket bag or more generally called a hold all are all pretty much the same thing) but the funny part is they're also generally more comfortable than most of the expensive branded options from lighting companies. An example of a suitable 2 head bag would be the Elinchrom 2 head soft location case, Elinchrom Protec poly bag, they're expensive but there's a ton of cheaper versions available (the older Prolinca's, Hensel do a reasonable one etc).
You can get a 2 head case in the £25-45 range
second hand and the hold alls as mentioned for under £20 new, this is my personal preference as it's very neat and easy to store in a home environment.
I get to lug around a lot of equipment and the most important thing for me is usually to get the weight onto my shoulders and never on the arms but it depends on the situation, if I knew I needed to drag it over a long distance with a decent surface I'd prefer a trolley case.