Ahri's a bit of a special case within the mages. She's got a little underwhelming damage, but makes up for that with hard cc and mobility. In the end, you'll need to outplay an opponent to win your lane. This drop in power is lessened with items, but just remember her power spikes are later than most.
As for the rest, recognizing power spikes through items is something I'm still learning. It's not gonna come easy, but it will in time. The best way to figure out who's stronger is to look at items and figure out where you and your opponent's strengths lie. If you've got only a Forbidden Idol
and an Amplifying Tome
while they've got a Blasting Wand
and a Faerie Charm
, you will loose trades due to them having more AP, and therefore more damage. However, you could play the lane in your favor over time through your higher mana sustain.
If you're going to try and join a fight, identify who your targets are first before you jump in. The primary targets are usually squishy carries in the back line. Once you know who you're trying to kill, figure out if you can make that happen. Do you have enough health to survive a skirmish? Is your team prepared to engage and lock them down for you?
All in all, knowing when to back off is more important, and also easier to recognize. Team's on low health? Back off. Low mana? Back off. Spells on CD? Back off for a little.
Now, as for last hitting, the best tip I can give is to just practice it. Doing it live will work, but some dedicated time will help immensely. The best method I've found is to practice it in custom games. Make a game, choose the champ you need practice with, pick a lane, and just farm. If you want, or when you feel ready, add a bot to simulate harassment from a laner. Try not to kill the bot, just harass them back until they have to back. Eventually, farming should come second nature to you, and you can practice wave control (freezing, pushing, etc.).
Hope this helps. Good luck!