Hi,
To scare someone off before they break in, try this dual beam IR sensor at the back of your block, (installed above where cats would walk).
You could wire this up via a relay to anything that makes noise, a car horn, a bell, lights, etc,
https://www.ebay.com.au/sch/i.html?_...rimeter+sensor
If you want the alarm inside the shed to work, which means you get a broken door before the alarm goes off, then I think PIR detectors of a good brand could be what you need.
They should be pet safe, so something small shouldn't set them off.
One alternative if you already have a house alarm is to get one with individual zones, and extend a zone to the shed, and have that zone turned on when you are not there, even if you are home in the rest of the house.
A good quality separate alarm for the shed, such as a Bosch solution 3000 sell for under $400 on ebay, and it isn't rocket science to install. (possibly plus the communication IP module and a roll of cable )
Alternatively, many of the new CCTV camera systems can set alarm outputs when there is motion detection in front of the cameras. I think you would have a few false alarms with that.
Disclaimer: I'm not security specialist in any way, so I'm sure someone here does this for a living and can give further advice.
Cheers,