It will always be a bit strange using a stranger at first. That is why as a babysitter I always meet families who contact me, prior to actually doing any babysitting. I find that helps a little, as we can have a chat over a coffee and the children can start to get to know me a little.
As an independent babysitter, I can only be in once place at a time, so families who go out often I find will use an agency as a backup for when I'm already booked. A plus point of using an agency is that they have many people they can call upon, but that is also a negative point - as it means a stranger coming to your home, rather than someone you and your children have got to know over time.
All my work now comes via parents who find me on the internet. I suppose that may be due to presenting myself in a professional way, giving a lot of details on my website (for example, you can read testimonials from other families), saying what childcare qualifications I have, CRB check and Ofsted registered etc. I find the more a parent knows, the happier they seem to be to meet me.
My tips for finding a babysitter these days are as follows:
Google Search using a phrase such as "babysitter in townname" - so if you were in Camberley, Surrey for example, you could use "babysitter in Camberley"
This will often bring up search results for local ad sites (such as vivastreet, gumtree) and also independent babysitters websites, like my own. Avoid the Sponsored Links, as those will typically be the agencies. Visit the websites, contact the advertisers on the local ad sites, and obtain as much information about them as you can before arranging for them to visit your home to meet your family.
From September 2008, Children's Information Service is supposed to be providing information about all kinds of childcare provision. They should be able to tell you about Registered Nannies and Registered Childminders working in your area. They may be able to offer occasional babysitting.
Try posting on your Mumsnet Local to see if any other parents in your area can suggest anyone suitable.
If you attend church, perhaps try placing an advert in the next church newsletter.
Hope that is of help.