Ditto! The trick is not the camera but catching the hamster when it's not moving
And good light.
Blurriness is caused by two things
1) Movement of the subject when the shutter speed is not fast enough to catch the action. To have a fast shutter speed you need good light levels.
2) Camera shake - when the shutter speed is slow the camera picks up the very slightest movement of your hands as you take the picture - which also shows as blurriness.
To allow more light in, you can change ISO settings. ISO 200 is standard and fine for outdoor daylight. Indoors in the evening you're more likely to need ISO 800 or 1600. The higher the ISO the higher the shutter speed you can have in low light - however higher ISO's also make the pictures have more "noise"/grain/pixellation.
I find having normal lamplight on in the evenings is enough light - you can either adjust the colour balance afterwards (to get rid of the orangey glow you get with tungsten indoor bulbs). Or you can set the camera to "tungsten lighting" which balances the light colour at the time of taking the photo.
Most cameras are incredibly clever these days. They have stabilisation built in to remove camera shake (if you turn it on!). They can do reasonably highISO settings without too much grain (but the lower the better for picture quality). They can be set to choose the highest possible shutter speed. And can takemultiple shots at one press to capture the best picture if something is moving.
I don't think an SLR is necessary. But a camera which changeable lenses can give a sharper picture due to lens quality and also enable you to set backgrounds to be out of focus with the right lens.
But some compact zoom cameras are extremely good as well. I wouldn't use a phone camera to get the best picture. They are ok in ideal conditions but not so good for specific settings or low lighting.