Fighting is generally dictated by Soft/Hard Attack, Organisation and Speed.
1) Soft/Hard Attack: SA for non-armored divisions, HA for Armored ones; (P.S. DH Combat is different from the one Hoi4 uses as the latter has Combat Width and Reinforcing Rate being other factors too. In DH there is no Width nor Reinforcement Rate)
2) Organisation: It is generally best you let your enemy attack first as defenders get some bonus in Org Loss during a battle (in fair ones at least). Don't expect to win a 1v1 with an infantry division against an armored one;
3) Speed: It is actually an indirect factor in a fight and more of a major factor in a war. Once you win a battle, it might take some more time for a division/army to reach the province you have attacked and, if the enemy has higher speed advantage (which is the case most of the times) then they can easily put a more organised unit compared to yours and send you back.
I have long learned to not give too much weight in fair fights in DH, I usually use the classic Human Wave tactic or even use tanks to create small encirclements.Defensive stances are usually the best in many situations, let your enemy waste their organisation and their manpower and then send them packing back home.