There is no way to "increase" storage on your iPad or iPhonoe, but you can clean up your device to make room for more photos and apps. Apple has made it easier to do this by adding a "Storage & iCloud Usage" section in the Settings app. This section will inform you how much storage you have used and how much you have left. Just navigate to the General section in the Settings app and tap the Storage & iCloud Usage tab to access the information. From here you can also tap the "Manage Storage" option to see a breakdown of which apps are taking up the most storage. You can "increase" your storage by deleting the most memory hogging apps or the apps you no longer use.
The biggest storage hog on most devices will be the Messages app. This is because it is storing all the images and videos people have sent you over the year. To clean out your Messages app navigate to an individual text thread and tap the "Details" button. If you scroll all the way down you will see all the photos and videos that person has sent you in the "Attachments" section. Tap and hold down on an image to bring up the "Copy | Delete | More..." options. Select "More" then tap each image and video you want to delete and hit the garbage can icon to delete them. Doing this for several text threads will clear a lot of space on your device.
Photos and videos stored on your Camera Roll can also take up a lot of space. To delete images navigate to the Photos app and tap the images you want to delete. Once open you can tap the garbage can icon to delete and individual photo or video. There are apps available in the App Store, such as Purrge, that can make this process quicker.
After you have deleted all the photos and images you want you must permanently remove them by navigating to Photos > Recently Deleted. Tap Select in the upper right hand corner and then tap Delete All.
You can learn even more ways to preserve space on your iPad or iPhone here.