There have been countless threads about Paladins, why they should fall and how they are often a problem at the game table. So I thought it would be worthwhile collecting ideas on how to play a Paladin as a constructive and collaborative member of the party. I'll start:
The Paladin should lead by example and not by instruction. The Paladin is a representative and champion of a deity. This means their code should be tweaked so that it represents the ideal of their god. It also prescribes how the Paladin should conduct themselves. It does not give the Paladin the authority to tell others how they should live their lives.
Respect others who have different beliefs and opinions. In the real world, different communities with different beliefs can live alongside each other harmoniously. This should also be true in the game world and it is certainly not a lawful and good activity to create discord between those communities. The Paladin should only act against an obscenity to their God. Turn the other cheek if it is merely an annoyance and rise above the provocation.
Revenge and vengeance are chaotic motivations, the Paladin must not act upon them. Delivering justice may require the same actions but the motivation and roleplay is different.
If you as a player are caught in an impossible moral quandary (e.g. kill an innocent to save others), ask the GM what your character would know to be the best options in their game world. Unless the GM is trying to screw you over, there will be at least one choice that keeps your Paladinhood intact. Note, your character may feel terrible for having made the choice and may spend the rest of their days seeking out the perpetrator and seeing justice done (note: not seeking revenge).
Closely related is the pointless self sacrifice. It is just that, pointless. If your GM has engineered that to be the only moral way out then consider taking it and leave the game. To keep the game running, one could argue that if your only options are self sacrifice or let the bad thing happen, then let the bad thing happen and seek to bring the perpetrator to justice, your God will be pleased that you haven't wasted your life and are able to continue being their champion.
When roleplaying a Paladin, try to act like the kindest, considerate, honourable and helpful version of you that you can be and you are unlikely to fall into the Lawful Stupid trap. If that version of you wouldn't do the lawful stupid thing then neither would the Paladin.