Health shrines and turrets are the only objectives on the map, and should dictate gameplay. Pushing towers is critical - a death in exchange for a tower or inhib is more than worth it.

Farm the health shrines. If you need health or mana and then take your outer relic (as opposed to taking it to deny it to your enemies), take 5 seconds and get the inner one too. You (and anyone who comes with) get a mini-back. This also maximizes the number of shrines that spawn under your control, thus maximizing your team health. Leaving the inner shrine for "when someone needs it" just means your team ultimately gets fewer shrines spawning and thus is at a disadvantage.

On a related note, denying enemy access to their shrines is very valuable. Steal them as much as possible - if you can get the enemy inner shrine while both their turrets are still alive it's major points for counting coup.