Welcome to the forum - I hope you´ll enjoy it here!
We all play the game in different ways, but what I did, and still do, is prioritize fast heroes first. I built teams of my three stars, I think six of them, so I could participate in wars. If going in underpowered, I stacked mono.
Of your bunch, I would give your three stars more love first. Do not worry too much about 4 stars, and in particular, 5 stars, as they require hard to get materials (luck depended) and a lot of resources and time to level. It´s best to build teams of three stars first, then four stars etc. Is my thoughts, at least
I would wait with emblemming and LBing anyone. The matchmaking in tournaments take your strongest heroes into consideration, so it´s best to spread out the power a bit, once you are there. Emblems and Limitbreak is for later on, I would say.
Of your bunch, I would focus:
Hawkmoon (healing),
Melia (crit, helps a lot in boss fights),
Brienne (great for attack up),
Gan Ju (mana control is key in many fights),
Carver (similar, mana control for the win),
Nashgar (great sniper), Azar (can´t have enough mana control!),
another copy of Valen (great for fast defense down),
perhaps Renfeld in hopes of getting his costume later, that´s really great too,
Berden (great for attack down),
then Tirburtus and Ptolemy.
Is what I would do (other than level the heroes you have the costume of as well, as Sarah2 said, that´s a great bonus to have!) I sadly found Graymane to be underwhelming, but I levelled him anyway, as I liked having a werewolf (take that into consideration when reading my advice )
I saved my gems for a 30 summon in Valhalla back when I was new - that greatly helped me along. But times have changed, and there are most likely other places that are better now, but the Costume Chamber is always great too, I hear. Maybe others can shed more light on that.
But for now you got a great bunch of heroes to share some love, while building your base, so enjoy growing
Good luck ahead and have fun, it´s the most important