For your budget, you can get a disco 1 but bear in mind they are nearly 20 years old at their newest and unlikely to be much under 100k miles. You could also get a disco 2, probably one of the earliest ones but again you may have to compromise on the mileage. A higher mileage is not an issue if the vehicle has been regularly serviced and looked after. Check service history too.
The disco 1 has a good chassis but the body suffers from rust, particularly the boot floor, front inner wings, body sills and rear wheel arches. You could get one that has had welding done, which would save a lot in terms of labour costs. The disco 2 has issues with rust on the rear chassis above the axle. Repair sections are available but again factor in cost of fitting. Also on the disco 2, check the electrics are working and the air suspension if fitted.
Both are quite comfortable for family use. I've had one of each and they are very versatile vehicles. As for getting one already modified for off-road use, it may already have had some use/abuse off road so check thoroughly. There are lots of discoveries about, particularly disco 2's so look at loads, test-drive and be prepared to keep looking. Decent disco 1's are starting to hold their value so a disco 2 may fit into your budget better. Off-road kit isn't cheap and a decent set of tyres will cost upwards of £500.
Others with their experiences will be along soon. Let us know how you get on.