First and foremost, clean the barrel! Use a crown saver or weedwacker line to pull some patches through and remove the rust preventive that often coats a new barrel.
That rifle is a clone of the venerable RWS 34, I don't care for the stock design myself, but to each their own! Probably the best thing you could do to improve it would be to install a Vortek kit.
http://vortekproducts.com/ourstore/HW97%20Air%20Rifle?product_id=240 I have a Vortek kit in my Beeman R9 and the shot cycle is amazingly smooth, just a nice thunk! There are vids out there on the process, you may need a spring compressor to safely disassemble and reassemble the rifle. Brace yourself, the kit is almost as much as the gun! I would shoot it a lot, figure out what pellet it likes best, and then decide if I wanted to tear down and put in a Vortek kit. If the rifle is twangy and jumpy now and that bothers you, then put in a kit, it will be a night and day difference.
John
PS-Make sure you have a solid mount, ideally with a stop pin to keep scope and mount from sliding back, a moving scope will make you crazy when you try to sight in!
Also, if your groups open up, make sure the stock screws are snug (don't overdo this, just nice and snug).