Characters in darkness without a light source or darkvision are blind

Darkness creates a heavily obscured area. Characters face Darkness outdoors at night (even most moonlit nights), within the confines of an unlit dungeon or a subterranean vault, or in an area of magical Darkness.

and

A heavily obscured area—such as Darkness, opaque fog, or dense foliage—blocks vision entirely. A creature effectively suffers from the Blinded condition (see Conditions ) when trying to see something in that area.

Blinded means is a condition in the game which says:

A blinded creature can’t see and automatically fails any ability check that requires sight. Attack rolls against the creature have advantage, and the creature’s Attack rolls have disadvantage.

Now that we have the groundwork laid, I'll apply this to your questions.

Moving in a dark environment

There are no rules that apply any kind of penalty when moving in darkness without a light source even in a forest or similar environment.

A DM might rule that it is difficult terrain:

Every foot of movement in difficult terrain costs 1 extra foot.

But there are no rules demanding or recommending this.

If at any time a character would need to make an ability check in the forest that depends on sight while blind they will automatically fail. This will definitely hamper their ability to perceive threats (enemies, dangerous plants) and environmental dangers (pits, tree roots, etc.) if there are any present.

Attacking in a dark environment (everybody is in darkness)

A creature fighting in the dark that cannot see has disadvantage to hit someone else and advantage to being hit by someone else. If you have two such creatures, the disadvantages and advantages both cancel out meaning that they only roll one d20 to hit like normal.

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa. In such a situation, you have neither advantage nor disadvantage.

Of course if you can't see the target to begin with, you are going to have to first guess their location before you can hit them.

When you attack a target that you can’t see, you have disadvantage on the attack roll. This is true whether you’re guessing the target’s location or you’re targeting a creature you can hear but not see. If the target isn’t in the location you targeted, you automatically miss, but the GM typically just says that the attack missed, not whether you guessed the target’s location correctly.

When a creature can’t see you, you have advantage on attack rolls against it. If you are hidden—both unseen and unheard—when you make an attack, you give away your location when the attack hits or misses.

Attacking in darkness (target is standing in light)

Note that the blind condition says:

A creature effectively suffers from the Blinded condition (see Conditions ) when trying to see something in that area.

So a creature standing in darkness can see the character in the light. Thus, they can target them normally no guessing or disadvantage. Additionally, since the other creature can't see the one in darkness, the creature has advantage on attacks against the creature in the light since:

When a creature can’t see you, you have advantage on attack rolls against it.