The cottage had a number of walking and cycling guides which we used to plan our route. For the first couple of rides we just bimbled around the local area to the North of where we were staying, taking random turns and then heading home when it was time.
First full route involved being dropped off in Caistor (to the North), doing a loop and then riding back to East Barkwith. Very hot, but great places to stop for lunch, ice creams, etc.
Second one was heading out East over the Wolds to Louth and back again. Yes the Wolds are hilly, but nothing that would stop a fit 7-year-old. Not saying he enjoyed it, but he got up and over. Once you are in Louth we did a circuit on some pretty flat countryside.
Last big ride of the holiday was the Water Rail route from Lincoln out to Bardney, Kirkstead and beyond. This is flat. Very, very, very flat. Possibly too flat. But the cafe at Bardney was very welcoming and the route is peppered with sculptures, making it fun and interesting.
Generally the roads were very quiet and trouble-free, but there again my comparison is Birmingham. There are some hidden gems in that part of the country (places to visit, places to eat, shops), but being in rural Lincolnshire they aren't exactly around every corner.