Not mistakes but general advice:
Before you place an ad be clear in your mind what kind of person you want, what things are non-negotiable, what you're prepared to be more flexible on, what duties you need them to do, what their schedule is, what you're paying. Be aware of what makes a candidate unsuitable (e.g. a nationality which can't get a visa).
Then write your ad giving as much detail as possible. That way you'll (hopefully) get au pairs with some idea of your job contacting you.
If you're planning to contact au pairs get familiar with the search tools on the website and narrow your selection as much as possible at first, the gradually widen your pool of potentials. If you start with too many you'll be overwhelmed!
Prepare a good set of written questions you can email people who contact you along with a more detailed description. If they respond to that, they're interested. If they don't, move on.
Set up phone calls/skype interviews with those who do respond and whose answers you like. Don't keep e-mailing forever as the au pairs will be in contact with other families too!
Get the DCs involved in the recruitment process. Other MNers say this is a great test of the au pair's English...
Have as many candidates on the go as you can keep track of. Only stop looking when you have a confirmed flight/train booking.
Become aware of your responsibilities to the au pair - if you pay more that the threshold for NI contributions you will need to register as an employer, for example.. Prepare a contract. Figure out what you're going to do about holiday (she is entitled to 5.6 weeks, which includes Bank Holidays).
Prepare a handbook for your au pair with a schedule of working hours and a list of jobs to be done. Bear in mind they'll need a fair amount of handholding the first few weeks. Leave a crib sheet of useful phrases/emergency vocab, go through contacting the emergency services in detail (short of actually ringing them!) - maybe do a role play if you're concerned they won't manage. Show them around the local area and give them a map. Try and find out about English classes in advance so your au pair can make some friends quickly.
I think the most common issue is when au pairs and families have different expectations. The only way to sort this out is to communicate clearly from the start and continue that communication. It's a good idea to sit down for a brief chat every day for at least the first week and set aside time for a weekly debrief thereafter. That way you can nip any problems in the bud and your au pair will hopefully feel comfortable enough to talk to you about any problems. When communicating bear in mind English is not their first language so be very clear and simple. Direct is often more effective than British politeness. 'Could you put the washing machine on?' is a directive to you and me but an au pair will hear it as the conditional. 'Please put the washing machine on' makes it much easier for them to understand your wishes!
You can use the advanced search to look through threads about au pairs in this section. There are some horror stories but don't let it put you off!
Good luck!