Do soldiers being trained add to your score and what's a good way to get a higher rank?

Yes they do count towards your score.

The best way to increase score would be to get a relic

Serious advice: I would recommend focusing on trying to gain as many conquers as you can rather than just using all your resources on building up an army (so try building/capturing outposts to make to gates to reach out to more areas) as eventually you will find yourself short of workers, having very high overheads and low income.

Resource outposts are a good way to increase your income and can increase your score slightly as well.