Maxing One Ability V.S. Balancing levels

its better to max one ability first then work the same on your next two skills.

Always important ofcourse that you should have access to all the 3 basic abilities in the first 3/4 levels

Entirely depends on the champion: Some, like Vladimir, are incredibly reliant on getting a single strong ability (Q, in Vlad's case) up as quickly as possible. Others (Morgana, for example) can max multiple abilities (Q or W, typically) based on whether they want better CC or more wave-clearing ability (Q for CC and W for wave clear).

Typically, however, champions are designed with 1-2 focus skills and 1-2 secondary skills. It's usually better to have a single powerful skill you can rely, especially since most champions have at least one "one-point wonder" skill (a skill that starts out strong and scales somewhat poorly with further levels). You put the most points into your best scaling, most reliable skill, one point in your one-point-wonder, and then max your third skill once your main skill is maxed out.

Exceptions include characters like Tristana: her Q is her power skill, but it's not very useful until you have a lot of AD to back it up, and her early-game resembles mage gameplay more than carry gameplay. A Tristana skill order might be W-E-E-W-Q-R-Q-Q-Q-Q: it gives you the highest early-game burst with 2 ranks in W and E, and then power levels Q once you have some damage items. Still, even there you could get away with simply one point in W (a one-point-wonder skill, potentially) and maxing Q and then E.

Depends on champ and match up. I play Fiddles a lot and my abilities will change with who I face. If I face a melee champ without an interrupt, I'm going to max drain and melt them when they try to fight. I also tend to max Q/W together in some lanes if I am going to have to rely on my ult to kill them (need them around longer so need some fear points lol). If I am against a counter or a long ranged champ, I tend to grab some early points in E so that I can harass. Well, I used to. With E nerfs that strategy is pretty pointless now.

always max 1 core ability first.

depends on the champions for example i like to delay maxing my ultimate on alistar since the damage reduction is the same at all levels. Plus his other spells have a long cdr so to be relevant in team fight you need to max qe to lower their cdr.