I have tires that are getting worn out. They were cheap tires when I bought the car. My question is, does anyone have a suggestion for new all season tires that will improve my grip in the snow and ice. If I have to I'll get a set of blizacks and store them until winter. Just wanted to get your opinions.

 

There are no good all season tires, they are 3 season tires.
Now they have All Weather Tires which are actually rated for snow. Are they as good as winter tires: no.
You may consider Hankook's All Weather Tire next set (Optimo 4S), see the video.

 

Would probably better off buying a good set of winter tyres and switching when necessary to be honest if you're that concerned

 

It all depends on where you live and how cold the winters are. If you live in Northern eastern USA you should be fine with All Weather Tires such as the above Hankook. If you live in Alberta you need winter tires.
It's all on your preference, budget, storage issues.
I would never buy an all season tire without a snow stamp on it though (which the above Hankooks have).
You never indicated where you live. To do so please read and do: https://www.mazdaforum.com/forum/new...-avatar-30905/ . I wish all newbies would do this first.