When should you start pushing?
At what point do you want to start pushing instead of just last hitting? Is there a reason to do it at all after any towers are down?
You never stop to try and last hit the minions, except when you're playing support; then you'll leave the last hitting to the carry you are with. But at times it's good to push minion waves to the enemy tower (either using your spells or starting to auto attack), since enemy towers can and will deny gold (and XP - only when your opponent isn't there) from your opponent. But you always have to look at the map when you are going to push, because overextending and dying to a gank is worse then letting your lane opponent farm.
Conversely, if the enemy is pushing his minions closer to your tower, you can start auto attacking the minions to get them down faster, but even while auto attacking you still need to watch minion hp so you can get the last hit.
Always last hit; you want the gold! Even when pushing a lane, time your attacks so you get the last hit.
I think your question could be rephrased as "When to push, when not to push?" In general, it's risky to push a lane because you're left in a vulnerable position. The two biggest reasons I push early are to heal with lifesteal, or because I'm beating my opponents in lane. It's risky, but an early tower break provides a nice boost for your team and frees you to float more.
Advantages to pushing:
- Tower damage
- Deny opponents exp/gold
- Healing with lifesteal
Disadvantages to pushing:
- Vulnerable position
- Uses resources
Some tips while pushing:
- Keep wards up in the river (you don't want to be ganked)
- Watch for MIAs
- Use care if an enemy champion has taunt. Taunt + turret = pain
tl;dr: Only push if healing via lifesteal or obviously winning, and then make sure you ward any potential gank spots.
There are a few times you should push.
- when you are close to an enemy tower and the enemy needs to recall
- when ad carry and last hitting goes well you want to be at their tower at around 8 to 9 minutes so you can recall and get your B.F sword
- around 15 minutes if you are winning and you can push down the enemy tower safely
After the 15 minutes push you should go to mid to get that tower down. Otherwise you should try to just last hit. Also if the enemy recalls and you're far away from their tower only last hit so they lose more experience.