How to add current date and time in the table so we know when a game score was logged?

You can use eosio::current_time_point() to get time in microseconds in smart contract.

#include <eosio/system.hpp> //include system header

uint32_t now() {
   return (uint32_t) (eosio::current_time_point().sec_since_epoch());
}

Add a scoreTime field in your score table

struct [[eosio::table]] table {
   uint32_t scoreTime; //add scoreTime in your table
   .
   .
   //other table fields
};

Add score time while inserting new score in your table.

table.emplace(get_self(), [&]( auto& row ) {
   row.score = 10;
   row.scoreTime = now();
   .
   .
   //other table fields
});