My iPhone screen is broken, what should I do?

I drop my phone a lot because I'm really clumsy and it just slips out of my hands, but today I dropped it and it decided to actually take some damage this time. I can't unlock it or do anything with it, but it still turns on, I can activate Siri (using the home button) and charge it and stuff. I just can't use the touchscreen. Will someone at the Apple Store be able to fix it? And if not, will I be able to recover all my saved stuff, because I have loads of music, notes and photos I would hate to lose. Should I go to the Apple Store anyway?
Here's a picture of the phone screen, if that helps. Sorry about the quality, all I have is a bad webcam

You could look online for phone repair stores near you. Firstly, you should back up your photos on Google Photos (It's the app with a pinwheel) and if you have an SD card, make sure everything is saved there. Then you can go to the repair shop and ask them to fix it. I don't know how much it would cost, but I'm thinking around $30; could be more, could be less.

A new screen cost around $50, i suggest you try to fix it yourself first, because a repair shop will cost you around $100-150 (never let apple repair your phones, they are expensive and your warranty is voided anyway)

get a good protective case

if your storage is not encrypted, then it will be safe unless the physical disk is broken
if it is encrypted, then you need that phone to read the information, else it will be very hard and very expensive to extract the data

I know that for androids you can use the device with a mouse, but i don't know if ios has this feature

The underlying layer of your screen, the actual capacitive touch layer has been damaged if it's not responding to touch. While the glass pannel (what most people call the "screen", incorrectly) is easy to repair, the capacitive layer is part of the LCD panel assembly and should be replaced by someone that actually knows what they are doing, ie Apple themselves.

Depending on your model of phone and the extent of the damage it might not be worth repairing and just replacing, but as for saving yourself, that's simple and you can do it yourself.

Hook the phone up to a PC, load up itunes and create a backup. Even if you do go the new device route, that backup image can be pushed down to the new phone intact.