Whether I play top, mid, or bottom, I want to get the most gold I can. Should I be a team player and help my team with ganks and such?
If I help my team I could be wasting my time getting gold. Should I just keep farming until my team needs me to group up?
This question is very dependent on your lane.
For example, if you're playing top lane you will often see players get a gold per five (gp5) item and sometimes even two. It is important to have map awareness and know when your team needs you.
In the top lane I would suggest last hitting and prioritizing gp5, while maintaining some semblance of when you should leave your lane to help. Ultimately three full minion waves is worth a champion kill. If you feel the need to leave to gank perhaps you should push your lane then leave and gank. This will also give you an opening in which the enemy may think you've just returned to base or are tower hugging.
Lanes like mid depend on your champion and ability to gank: a good mid laner will know when his teammates need help and will gank accordingly. Again keep in mind that minions are gold and staying away from lane too long is experience and gold lost. Map awareness is key and without knowing when and where you should gank, you're probably already a step behind.
As jungler, get a gp5 and gank accordingly. Map awareness is key and knowing when you can dive a tower or sneak into lane from your tower will make all the difference. Try not to lose important jungle minions as they are worth the most gold. If you're being counter jungled, ward and ask for teammates to help with intercepting pesky jungle invasions.
Bot lane / support is the most crucial gold farm in the game. I would say camp this lane and really focus on letting your attack damage carry (ADC) get minions. If you leave the lane he will probably die or be ganked. Wards are essential to letting him farm and if he dies, his gold farm dies with him, effectively preventing a strong late game carry. You and your ADC should only leave lane once the tower is pushed or you're defending dragon. But your role as support is to support. Build gp5 galore and aura items as you see fit.
Hope this is helpful. :)
If you are laning, the most effective way for getting more gold by far is last-hitting. There are many other methods out there, but they're most likely not the ones you want to hear.
You may hear of last-hitting a lot. Basically, last-hitting consists of waiting around until the minion has a health low enough until you can kill them with one hit. This increases your chances of getting the money from a minion kill, or "creep steal" by maybe 40-70%.
Good last-hitters can get 100-200 CS in 20 minutes. That's a lot of gold! Combining that with the gold you gain periodically and whether you're slaying a siege minion,
You can probably afford an Infinity Edge 20 minutes into the game and then some. This is considering you haven't killed a single champion yet.
With this, you can branch out on either being a team player or just keeping to farm, as the difference to you personally will be considerably more miniscule. Decide for yourself, but you'll be kicking ass either way.
Learning to last-hit takes time, but if you do it right you can become an extremely formidable force. Every good player should know it.
Best of luck. This will be helpful:
EDIT: The record was set by CLG.EU (my personally favorite team) in the Season 2 LoL World Championships of 300 CS in 23 minutes. That would be approximately 6400 gold.... from farming.
If you watch some of the top players, regardless of lane (exception is support), you will notice that they always work very hard to last-hit minions. I don't remember the exact math but it only takes somewhere around 10-15 extra minion kills to achieve the extra gold equivalent of a kill. So in my opinion you should always having last-hitting in mind.
As a mid player you can also grab more gold by clearing the jungle camps near to you. Only do this in your own jungle if you see that your jungler is not anywhere close to you and so won't miss having wraiths or wolves down for a minute or so. Bottom and top lane can also do this by farming double golems and wolves depending on color.
You can also counter-jungle the enemy wraiths as a mid player but usually only if you have the minion wave pushed up to your enemy tower and you know where the enemy jungler is. You can also counter-jungler as a top laner if you are really dominating your opponent.
Helping your team with ganks is your job as a jungler so this is a no-brainer. But as a mid player some champs are excellent at ganking. It is very common to push your wave up to the enemy tower and then go roam to another lane to get a gank. I do this often with Veigar. But in my experience you have to be very careful doing this because if it doesn't work out you might lose out on CS (last hits) or even worse you might get killed.
