It depends how much you've really got to spend.
You go through companies/stockbrokers when you buy stocks and shares (unless you've got loads to invest) - so you tell them who you want to invest with and they charge you a transaction fee (usually about £8 in the UK). Unless you're intending to invest a couple of hundred at least, it's actually pretty unlikely that you'd make much profit in the short term if you even manage to cover the £8...
For example - say you're only wanting to invest £50, the stocks of the company you've invested in would have to go up 16% just for you to cover the £8.
That's just something to bear in mind - I know it sounds insignificant, but I'm just trying to illustrate how pointless it is unless you have at least a couple of hundred to invest.
Now then, onto who to invest with:
You get high risk and low risk shares. Low risk would be the bigger companies like Sainsburys, McDonalds, etc... Who you know will grow and make you money in the long run, but you know they're never going to come up with a world changing business plan that's going to double your money over night, but you would expect to make several % in a year which still would be better than a bank these days.
Smaller higher risk companies however, (a Pharmaceuticals for example) - if they discover a new drug, they can get a 20 year patent for producing that drug, be the only company legally allowed to product and sell it, and so effectively make you loads in a short space of time. However, if you invest in company A, and company B discovers the drug and gets the patent, then you don't make anything, and could easily end up losing all of your money. Think of it like a 1/20 chance if you go this way.
How much money do you actually have to invest?
As appealing as high risk investments sound, if you do it with a few hundred quid - maybe even a couple of thousand, you'll most likely end up losing your money. The point of trading is to spread your money out across multiple companies so if one fails, you've not lost everything. But, as I said in my first paragraph, due to fees, you can't really do this unless you have several thousand pounds to invest.
As for what you say in the OP about banks offering it...that's not entirely true. The banks that will offer it will give you say 3/4% a year and then pool everyone's money together and go and invest themselves. Chances are they'll do this in low-medium risk companies so you're unlikely to lose your money, but if the bank makes 10% on your money for example, chances are you'd only see half of that as the bank will make profit on it too. This isn't really trading stocks however - it's more like a low risk bank account, where you may see more % interest than you would in an ordinary bank account.
You also get government guilts, which is similar to the bank account investing the money, but the government back up all the money you give them so you can't ever lose the money you invest (you may not make money, but at least your safe).
I sense you're just wanting to do this for a bit of a hobby yes? In which case, I doubt the bank accounts or government bonds would apply to you as you don't really do anything with it.
As I've said a few times though, if you're just wanting to do this to have a bit of fun, and only have a couple of hundred £ to invest, then invest in lower risk companies - you might make more than if it were in a bank, but if it's only a few hundred pounds, then that probably won't matter. Don't invest in higher risk, because you'll just end up losing all your money, and tbh, with the amount of money you're likely to have, you'd probably have better odds playing roulette.
This is probably the first time a degree in Economics has come in useful on these forums
