I find this to be a common challenge when learning Newton's laws. Newton uses a very precise meaning of force. This is close-but-not-quite the same as your intuition, so it takes a little while to get used to this. Netwon's law states that if the forces on an object are equal, then the object does not accelerate. This means it will keep a constant velocity.
Where your intuition is likely faltering is that many of the most natural forces we deal with are not constant. They change with position. For example, you'll learn a spring pulls with more force if you stretch it further. In these sorts of cases, you'll typically see unequal forces appear and you'll see the changes in velocity that you and I both intuitively expect to happen.
If I may offer an example you can try yourself, find a nice flat section of road with no cars (like a parking lot) and a friend/parent/etc. Have them sit in the driver's seat (for safety) and put the car in neutral. The car wont move. Now, go to the rear bumper and start pushing it. No surprise, the car will start to accelerate. The forces on it are not in balance.
Get it up to a reasonable speed, and then stop pushing. Now we should all agree that there are no forces on the car acting in the direction of motion (just gravity and the normal force from the ground... which goes up and down rather than forward and backwards).
What do you expect to happen? Do you expect the car to suddenly halt because you stopped pushing it? No, you'll find the car happily keeps moving in a straight line, at whatever velocity you pushed it. Remember that friend I mentioned was in the driver's seat? Yeah... this is where they'll probably need to apply the brakes.
Now you will notice the car slows down. Why? Because the friction of the bearings on the axles isn't 100% perfect. But they should be close enough to perfect to get you some gut intuition as to what would happen if the axles were indeed perfect. Without their friction, the car would just keep moving forever (or until it hits a wall. Remember your friend in the driver's seat?) The makers of cars have been very interested in minimizing that friction because lower friction means better gas mileage. So they've done a pretty darn decent job of it.