Hey there!
When I was levelling Zane, I stuck to the blue tree for the first 26 skill points or so. I didn’t equip a second Action Skill until after hitting the capstone, as grenades were pretty helpful until I had more points to throw around later.
As far as the blue tree goes, even before the recent buff, I found SNTNL worked well enough for levelling through normal mode. Plus, the blue tree itself has a lot of skills I’d consider to be very solid basis for most Zane builds too.
Violent Momentum is a great damage boost just for walking while shooting. Cool Hand may not seem like much, but you reload at the end of every mag, and it all adds up. Death Follows Close significantly enhances all your Kill Skills for just one point, and Salvation adds some great burst heal too. Other than that, and later down the tree, Playing Dirty is fantastic for high damage, low rate of fire weapons; just grab a class mod to boost the percentage. If you prefer high rate of fire weapons, run with Violent Violence instead, as this can really bump your damage output via rate of fire. Once you hit the capstone, you’ve got all your Kill Skills on demand, and that’s a huge power boost in itself.
That said, you can always mix and match from other trees if you find things aren’t working for you. Skills like Synchronicity (Digi-Clone tree) are a solid early damage buff, and Ready For Action (Barrier tree) can help with your shields and survivability.
After maxing the blue tree, I focused on Cooldown, as I wanted my action skills activating as much as possible to trigger kill skill bursts. I also switched my Action Skills over to Barrier/Digi-Clone at this point, to make use of their low cooldowns. Adrenaline (Barrier tree) is the place to start beefing your cooldown, and I worked my way up to Confident Competence (Barrier tree) for another decent damage buff.
I found this approach worked great with shotguns for the majority of the game, though you could do just as well with SMGs or Assault Rifles if built correctly too.
At any rate, there’s certainly many different ways to level, and I’m sure others will give your their methods too. That was how I did it however, and it got me through to level 50 well enough.
Good luck!