Visiting other breeder's ranches
Clicking menu then contacts will bring you to a variety of options for visiting other people's ranches. When you visit someone you have the options to pet and/or give their monsters presents. Each monster petted will give you one monster medal which can be spent in the store at the gacha machines, and the pet will train harder the next time his breeder sends him to the task. If you give a monster a present then you will receive no less than five monster medals, though the number goes up with the rarity of the present given, and the fame of the monster goes up.
When you visit someone’s ranch click on 'Breeder information' then click on 'Neighbor request' on the top of the page to ask to be their neighbor. If they accept then they will be added to your 'Neighbors' page, which is a list of people whose monsters you are friends with and should pet once a day.
The max number of neighbors is level 100. You can get 100 monster medals per day just by petting neighbors. (This total does not include monster medals gained for giving a monster a present.) A good player always pets all the neighbors.
Tips in picking helpful neighbors
First thing that you should look at is the last login date. If it isn't in the last two days then don't bother, they probably dropped the game and are not likely to log back in to accept your neighbor request.
Second check their rank. You can combine with your neighbors monsters, therefore you want your neighbors monsters to be the same level as your monsters (or higher), otherwise the newly combined monster will be a lower level than it should be. Try to make sure that your neighbors are around the same breeder rank as you are.
Third, when on an actual neighbors page look at their history (a book icon in the left side of the screen) and see who the last person was to pet their monsters. Make sure that the petting was within the last two days, if it was then visit that person's ranch. This person is known to pet other's people's monsters so they are more likely to pet your monsters, giving you extra stat boosts!
Through the history book you can also see who has given a neighbors pets a present recently and choose them to be your neighbor. This is a little iffy. Some breeders only give presents to people who gave them presents while others give presents to cool looking monsters. Since presents cannot be bought through the normal store there are a limited amount to give out, therefore you will not get one often even if the neighbor is known to give our presents. To be efficient one should give a present to a younger aged monster so that they can use their fame within battles throughout their whole carrier.
Once you have 50 neighbors you should go through your list once a week or so to see who isn't playing anymore. You can remove them from your list and add someone who is more likely to help you with your pets.