Hello there, my first print with the Anet A8 was just perfect. For your problem, I think we will need some more details. What material are you using? The last time I had such a problem was the filament diameter not correctly setup in Cura.
This could be because of several reasons. I would do the following checklist:
Check that the Z rods are fixed in the holes in the top of the Anet, but should not lie on the stepper driver axis. The couplers of the Z rod should flex a bit if you press the on the stepper motors.
Check all belts if they are really tight. They must be REALLY thight so there is no play.
Check your temperatures. I print PLA with 195-200 degree and PETG with 220-230 degree (depending on brand and color).
Check print speed. Start a little slower, you can always increase the speed in later prints. Start with 20mm/s.
And also, here is a very helpful troubleshooting guide for all mistakes everybody will do at some day ;)…
Unfortunately the stock Anet needs several improvements. Hint: also check the heatbed connector, this will also break at some time.
About your speed question: Start printing with 20mm/s for perimeter, walls and infill. The stock Anet is defintely NOT an high speed printer. 20mm/s is really slow, but this way you can prove that speed is not the problem and eliminate all other problems. After that you can increase speed by 5mm/s for infill for example and experiment with different speeds. I added a bowden system to my Anet to be able to print reliable (!) at higher speeds.
I only had to reduce the first layer speed. Everything else at 60mm/s is very good on Stock Anet. It seems to depend many non-understandable factors! I need to reduce the speed if I want really really good and precise prints.