Well there is Mablethorpe for the traditional seaside experience. Walk along the beach to the seal sanctuary and get the sand train back (it's cheap but fun)
For an unspoilt beach day out go to Huttoft car terrace. You can drive up to the beach. There is nothing there (a very cold and old toilet block) but the beach is fantastic. Sand, sea and sky! An ice cream van will probs be there at some point during the day, but you will definitely need to pack a picnic and take your own buckets, spaces etc. It's the east coast - a wind break will be invaluable at some point during your holiday!
There is the North Sea Observstory at Chaoel St Leonard's - it's dog friendly. They do lovely cafe food.
You can check out the Lincolnshire Coastal Ranger on Facebook - there is an events page which will be updated with events happening in and around half term.
Lincoln is lovely - it's my favourite city. The castle is good fun. The prison chapel is eerie but the kids will love it. The museum of Lincilnshire Life is worth a visit too and only a short walk from the castle. A walk up Steep Hill is a must! Much nicer quirkier shops at the top of the hill. The Pinento tea room is nice - don't be put off by going through a shop to get to it. Fab veggie food and lovely cake. There is the Usher Gallery and The Collection (museum) quite close to the castle/cathedral - handy for if it rains! At the bottom end of the town is a shopping centre and Stokes Tea Room - their coffee is fab. You can get it at The Collection as well but the tea rooms in the town centre are in a half timbered building.
Tattersall Castle is a good day out - climb to the top for done fab views. Nearby is RAF Coningsby so quite often you'll see a flypast. Near Tattersall is a water park with huge inflatables - great for children (they will see it from Tattersall Castle so be prepared!!)
There's probably loads more to do, there's Skegness - Gibraltar point and nature reserve, the Lincolnshire Wolds if you like walking. An aquarium in Skegness.
I'm not keen on the east coast in the summer months - too busy! So stick to inland activities. But I'm sure you'll find plenty to do