Fir4stly, a three man tent sleeps 1-2. A family of three needs a 5 man tent- the manufacturers follow the plus 2 equation so buyers must as well.
What you need to do is set a budget, ensure that any tent you like has a minimum HH of 2000 (waterproofing level) and you need to decide if yu are happy to not be able to stand in the tent or if you need height.
A good way around it is to look at second hand on ebay. you can check tents out on the uk campsite website. Often you can get tents cheaply by finding a pick up only ione then asking if courier pickup arranged by you (someone like parcel2go) is ok
Do you know anyone that has a tent you can borrow? Doesn't matter if it's too big as it would just be to see if you actually like camping or not. I must admit whenever we go I always nose around the campsite and find that most people are more than happy to show you around their tent and talk about the pros and cons. Great why of finding out what's suitable without having to buy anything.
Whereabouts are you? There might be some big tent showrooms in your area...