TL;DR

Many of the effects of haste and slow negate each other, with the following exceptions:

• Use one action or bonus action, and one action from the extra actions on the haste action list.
• Have advantage on Dex Saving throws with a -2 Penalty.
• Can only attack once per round, regardless of skills, abilities, or magic items.

There are some specific effects, but they're unambiguous so I'm leaving them out of this description.

Let's examine both in detail:

Haste(Pg. 250 PHB)
Choose a willing creature that you can see within range. Until the spell ends, the target's speed is doubled, it gains a +2 bonus to AC, it has advantage on Dexterity saving throws, and it gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon Attack only), Dash, Disengage, Hide, or Use an Object action. When the spell ends, the target can't move or take actions until after its next turn, as a wave of lethargy sweeps over it.

...

Slow(Pg.277 PHB)
You alter time around up to six creatures of your choice in a 40-foot cube within range. Each target must succeed on a Wisdom saving throw or be affected by this spell for the Duration.

An affected target's speed is halved, it takes a -2 penalty to AC and Dexterity saving throws, and it can't use reactions. On its turn, it can use either an action or a Bonus Action, not both. Regardless of the creature's Abilities or magic items, it can't make more than one melee or ranged Attack during its turn.

If the creature attempts to Cast a Spell with a Casting Time of 1 action, roll a d20. On an 11 or higher, the spell doesn't take effect until the creature's next turn, and the creature must use its action on that turn to complete the spell. If it can't, the spell is wasted.

A creature affected by this spell makes another Wisdom saving throw at the end of its turn. On a successful save, the effect ends for it.

So we can see that many of the effects are cancelled for both spells, specifically:

• Movement speed changes
• AC Changes

This leaves two components that we can break down:

1. What actions can and can't be taken
2. Non-cancelled spell effects (Out of question focus, so I'll leave this alone)

With haste, the additional actions can only be:

1. Attack (One weapon attack only)
2. Dash
3. Disengage
4. Hide
5. Use an Object

With slow, you're restricted from doing certain actions:

1. Making more than one attack per round
2. Only take an action or bonus action, not both.

From this we can gather that being both hasted and slowed will basically cancel each other out with a few restrictions:

• You will only be able to make one attack action, regardless of abilities or magic items.
• You have the ability to take either an action or bonus action, and an action from the list of extra actions haste allows, however this is restricted to one attack per round, regardless of spell effect.
• A character with both might Attack, move, and disengage. Or Attack, move, and use an object. But never attack more than once.

Addendum
Note that you have advantage on Dex Saving throws due to haste, but -2 on those rolls due to slow.