I've been asked this question a few times by various people and as it's becoming a more frequent occurrence I thought I'd just post the process here on my blog.
The first thing you have to do is sign up to an exchange like Kraken.com (other exchanges are available). However, because demand is so huge it will take a long time - up to 4 to 5 weeks - for them to complete their due diligence on you. This involves them reviewing a copy of a recent bank statement and your passport which you must upload after you register.
When this verification process is complete you can send them some money and use it to buy your preferred crypto$.
However, if you leave your crypto$ on an exchange (a "warm" wallet) it could be hacked; people lost a lot of bitcoin when the mtgox exchange was hacked.
You can also transfer your crypto$ to a cold wallet on your pc. For example, "bitcoin core" for bitcoin (the wallets vary for each crypto$) and "mist" for ether. If you go down this route however you will have to download the blockchain for each crypto$ on your pc and keep it up to date if you want to send crypto$ anywhere quickly. Start now as it takes a very very very long time to download!
Finally, there are also little hardware gadgets like the Ledger nano s that you can use to store major crypto$ on that may even be more secure than your PC. However, I don't know much about these yet. [Edit: I just bought one. I'll let you know how I get on :)]
As always, please do not treat this post as financial advice. Crypto$ is very volatile and you could lose all of your investment!