Scopes do not like to be suspended from a single mounting point. That adds additional stress to the scope tube and allows flexing, it is after all thin walled aluminum tubing. It is better if the scope is mounted at two spots that are not close together to give it a more stable position. Look for a mounting system that will place the scope where it needs to be to give you a full field of view and be fully supported.
Another thing, is that a rimfire scope? If it is not a rimfire scope that means the parallax setting is for 100+ yards and will give you fits at under 100 yards. That is important as parallax can change your point of impact just by changing your check weld on the rifle. Unless you align your eye in the exact same place to shoot the scope it will shift the group location. If you move during the group and then go back to finish it it will be worse. You can find decent moderately priced variable power scopes set for rimfire where the parallax is set for 50 to 100 yards. Do NOT mess with the power setting until you get the rifle and scope dialed in. Cheaper scopes will change the point of impact as you change the scope power. Once you are set then you can do a repeatability test for the scope click settings. That is a more advanced practice than what you are ready for at this time.
Check the trigger for how much effort it takes to make the gun go bang. You want between 3 to 5.5 lbs trigger pull for a standard style field trigger. A custom target trigger can go into ounces of pull but you get that after you have become proficient at shooting.
Don't sweat the brand or velocity of ammo just yet. Practice first until you get consistent ROUND groups. When I say consistent, that means in the same position of the target for the same aiming point. Once you and the rifle are set then you can look for the exact ammo for precision. It may well turn out to be the cheapest stuff at Walmart. Rimfire rifles can be really strange about what ammo they really like.
