Arequipa - you can get all of what you're looking for, except for close to the beach - it's about 3 hours away. The weather is good, although it can be rainy in February and March. It's a lovely city - easier to get around then in Lima, great food, lots of areas to live in, reasonable cost. Places for your children to go to school and learn Spanish. Houses might be hard to find, but you can find apartments with a park nearby for the children to play in. biking isn't really recommended as a means of transportation, but walking is common and buses and taxis are very inexpensive.
Ask to join Expats Arequipa on Facebook: https://www.facebook.com/groups/arequipaexpats/
- the moderator asks you to introduce yourself to have you join the group.
Members will be able to provide information on specific questions.
Good luck, Arequipa is a lovely city .