(Not sure your starting knowledge, so I've try to give a quick brief, other might be able to point anything I've missed that you should consider).
Investing in funds in the uk can usually be done via a stocks and shares platform (sometimes called a supermarket) online. Most have either standard investing accounts or 'Stocks & Shares ISA' accounts that count as part of your ISA tax-free savings subscription for the year, and means can be money can be transferred between existing ISAs without being classed as part of your current yearly tax-free subscription.
(in the uk atleast) investment platforms usually charge a monthly or fixed admin fee for holding your accounts. This usually within the region of 0.25%-0.40% of your current account balance and is a separate charge. i.e. not taken from your current savings amount. So the more you have in the higher the fees, some providers cap their fees at set amounts per year/month, so it's could be worth while when you eventually have a large amount saved up.
This site gives a good run down and comparison of the popular fund platforms and things to consider when picking.
http://www.thisismoney.co.uk/money/...ck-best-cheapest-investment-Isa-platform.html
People find they might prefer one over another due to the help services and advice that a platform gives, Some might only specialise in Funds, or shares & ETFs so it's worth looking into what you would be wanting to trade in the future. Your bank might often provide a funds platform account, however they usually charge slightly higher monthly percentage Admin fee, and might not have access to the range of funds that independent platform might have.
The minimum order amount is usually £100 when buying a into individual fund, although some offer lower minimums if you set up your investment as a regular monthly plan.
Personally I'm with Charles Stanley Direct for my (index & active) investments as I liked their platform, they've been friendly with any queries I've had and fees were small compared with others for the amount I currently have saved. I might move once the monthly fees eventually overtake the capped fees of some of the others, personally any fees paid is small compared with (ideally) how much your investments are growing.
Overall, spend a bit of time researching and having a look what you'll have access to before plunging in, you'll feel more confident when you do.