Are you shooting pets indoors or outdoors?
What are you using for lighting?
How long have you been working with the lenses you currently have?
There's a tendency for newer shooters to start looking for "better" lenses when their shots aren't turning out the way they'd hoped they would; The way they see really great shots online from others. And the first thing that gets the blame is "kit lenses".
The truth is that kit lenses are actually quite good. The problem is that ALL lenses need the person behind them to learn how to best work them, AND the subjects, AND the backgrounds, AND the lighting, ALL together, in order to MAKE it all come together. "Better lenses" aren't a magic bullet to better photos. To get great photos, you need to get understanding and control over EVERYTHING.
Are you already shooting with your aperture wide open to get a shallow DOF? Because that's what those "better" lenses allow you to do. If you're not opening up your current lenses all the way, getting lenses that open up even more isn't going to fix it.
Are you paying attention to the backgrounds and doing what you can to get them as far behind the pets as possible? This, combined with opening up the aperture so that the iris is round and not jagged from the aperture blades is one of the things you really need to pay attention to in order to get that great bokeh you want.
Are you using off camera flash for lighting? Softboxes? Umbrellas? Reflectors? Scrims? On camera flash? Sun coming through windows? Neutral density filters? Understanding light and shadow and how to make the best use of them in different situations is what separates the snapshots from the photographs. Full control over your lighting also allows the control you need to get the aperture and shutter speeds you need for the round bokeh and more blurred backgrounds you're looking for.
What do you do when it's so bright that you can't open up your aperture to get the blurry background and round bokeh you want, even with the ISO as low as possible? Do you just crank down the aperture and then wonder why the background looks too sharp, not the buttery look you want? What do you think you'll do in that same situation with a "better" lens that allows you to open it up even more, when you simply CAN'T, unless you have some other way to get a handle on the light? That's where that ND filter comes in, by the way. Do you have one?
Your kit lenses, in most cases, can do that stuff you're looking to do, IF you HAVE control over the various aspects of your photography, and THAT is the difference between an actual photographer and just another person with a fancy camera who knows how to aim it and push the button.
My advice is: MASTER your kit lenses first. Squeeze every last drop of potential out of them first. THEN, when you FULLY understand EXACTLY how (if) they're limiting you, reach out and buy a "better" lens.