What happens to our force when we walk on ice?
the force of friction appears when there are two contact surfaces that interact against each other
If you by "interact against" mean "try to slide over", then correct.
As long as the threshold of this friction is not surpassed the Force of friction will adjust itself to any existing opposite force, balancing it.
Indeed. And to be accurate, every mention of friction here refers to static friction (as opposed to kinetic friction), which is what happens when sliding is prevented (when something tries to slide but doesn't).
If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.
Yes, although a bit combersome sentence. A surface does not "have friction". Rather, it has a roughness so that friction can appear when sliding against another surface wants to begin.
For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...
When you apply a backwards force with your foot - we can call it a stepping force, if we will - then as per Newton's 3rd law the ground responds with an equal but opposite forwards static friction. I call it a stepping force but there is no typical official name for it as a whole, as far as I'm aware. Depending on scenario you might also call it thrust or traction or the like as mentioned in a comment.
If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip.
What you mean here is that there is an upper maximum limit to static friction. But note, your stepping force cannot be bigger than this limit. Your stepping force can only exist if an equal but opposite static friction force also exists (again, this is Newton's 3rd law). If the ground cannot respond with an equal static friction force then it let's go, meaning the static friction force disappears.
Then you don't have to apply any larger stepping force since your foot is just slipping and sliding and not feeling any resistance to apply force against.
But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.
As described, it doesn't "go" anywhere because you can't apply a force against nothing. When you punch into empty air then you aren't applying any force; likewise when you step backwards you can only apply a force equal to whichever resistance your feet meet.
Instead your foot just slips and slides backwards. Kinetic friction would takes over now, and then your stepping force equals that one instead. But if no other force take over - if there is no other resistance against your foot now - then you will just be "running on the spot" and will never move. As if running on ice, or if you are running while hanging in free space in a space station.
This can in no way accelerate you forwards. Only a forwards-pointing force could do that. This is the reason that it is not your step which causes you to walk forward, it is indeed the friction from the ground.
What needs to be incorporated into this picture, is that a walking human in this context cannot be approximated by a point-like object - the approximation that we commonly make in introductory mechanics. Walking implies forces applied between different parts of the body, notable between the legs and the torso, as well as the forces acting between the feet and the ground, and the gravity.
Thus, the force that makes us move forward is not the friction force, but the force exerted by our muscles, making the parts of our body move in respect to our foot, held in place by a friction force. The foot meanwhile remains static - they do not accelerate.
The ankle exerts a force on the foot, and the foot exerts a force on the ground, which is counteracted by the friction force. If the static friction force is not able to hold the foot in place, as it happens when walking on ice, the foot will slip and accelerated by the ankle force.
I will try to explain some of this but will need to dissect your statements.
If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.
To be clear, the road does not have friction. Friction exists between the road and your foot. It is a property of both materials that are in contact. If you look up coefficients of friction in a book you will not find the friction of concrete, but you will find different numbers for concrete-wood, concrete-concrete, concrete-rubber, etc.
For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...
This is a common problem conceptually, especially when you consider Newton's 3rd law and the fact that the Earth is not an infinite ideal constraint. You are in control of your foot and can choose to slide it along the surface of any other object. There are two types of friction and our understanding of them is empirical, though we have equations describing them. Static friction is what allows you to walk and this force will balance the component of force you exert on the ground that is tangent to the ground, as long as there is no relative acceleration. In contrast, pushing down does nothing w/r to activating friction. There is a limit to how much the static friction force can grow and that is equal to $\mu_s ||N||$, where $||N||$ is the magnitude of the normal force, the force perpendicular to the interface between the two touching objects. In the example of you walking on the ground this would be due to the earth pushing you up, and by the 3rd Law you pushing the earth down. Like friction, the normal force is understood empirically. The force of gravity between you and the earth draws you and the earth together. Once you and the earth touch it is the normal force that prevents you from merging with or passing through the earth. And it is basically due to the strength of the molecular bonds in each solid. In contrast, if you and the earth were in a liquid or gaseous state you would pass through each other and mix into something. Going back to friction, once the sideways force you exert on the ground grows beyond the limit static friction is broken and slipping occurs. The two touching bodies still have a degree of roughness and there is a force called kinetic friction that takes over. Like the normal force both types of friction can be understood at the atomic level to be the result of the strength of molecular bonding between atoms in the surface, as well as the variations in height of each surface at a micro level.
If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip. But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.
First of all contact is needed for you to exert force on the earth so once you've "slipped" at that moment the force is likely gone. If you maintain contact then the static force is replaced by the kinetic force of friction and the force you are exerting may have changed but is still there. I think the part you are trying to get your head wrapped around is that you feel like you need the ground to push back to exert a force in the first place and that is true. If you were standing on a frictionless surface (like ice) the you could not exert a sideways force on the ground. You could however move you foot backwards relative to your body. This would cause the elements of your body to redistribute in order to keep your center of mass fixed. This is a requirement of nature, a conservation law. As you throw your foot back hoping to grip the floor your torso moves forward. Pieces of you accelerate but in opposite directions. In the case of having a grip then loosing it because you pushed too hard what has happened is that you have changed the surface, and the relationship between you and the earth, to such a degree that it is now a different problem. Slipping absolutely could mean accelerating depending on the state of the body. To understand this you need to analyze the states in detail for a given problem. When a ball rolls on the ground it maintains static contact with the ground at the point that is instantaneously touching it. Thus friction makes it roll. This is sometimes called rolling friction in text books. Under these circumstances the ball will roll horizontally at a constant speed. If at some point slipping occurs then that means the point of contact is lost and there is relative motion between the contact surface. This will in fact create a horizontal deceleration (slowing down) of the center of mass of the ball. Similar statements could be made about your foot.