I hate everything to do with River so i hope they do drop her by the wayside.
River Song epitomised everything that was wrong with Moffatt’s showrunning: create a secondary feisty female character, make her the plot focus, give her an elaborately portentous but ill-thought out destiny, base the series arc around her, shoehorn in some contrived and leaden sexual tension, ultimately fudge the plot details and wrap it up in a hurried and unsatisfying blur, then repeat the formula next series. Also watching Alex Kingston smirking and posturing was like eating tinfoil.
I should also have added: neglect non-arc stories as inessential filler in thrall to the Grand Design, not allowing potentially interesting characters or ideas to develop organically.
I pretty much agree with most things mentioned above. I hope the new season will be, for lack of a better term, more down-to-earth. Not in the literal sense, but just less of the vague hand motions “whole universe and time itself at stake” camera zooms in on Doctor’s sweating, serious face “. . . unless one man, the loneliest man, the last of his kind, blahblahblah, fancy special effects, end of show.”
Not to be a traditionalist, but originally the Doctor really wasn’t all that special. He was basically a Gallifreyan college dropout who stole a TARDIS and took off into the depths of the universe because he found the normal Time Lord life boring. IIRC, he was actually escaping some minor criminal charges too. Sure, the whole thing about Gallifrey ending up destroyed changes that carefree dynamic, but the ultra-serious “most important man in the multiverse” thing is getting old fast. As is the romantic subtext with assistants. For that last one, at least, I hope the older actor will help. Everything else, I think, depends more on Moffat than anyone else. Despite the fact that I couldn’t stand Matt Smith, I believe he was a perfectly capable actor who was handed cheesy scripts.
In terms of plot developments, I think a child on board the TARDIS would be awesome (provided they can act, of course). Having something like the mother and daughter team of the Hartnell days could work well. (Or father - hey, it’s the 21st century!)
Of course, I’ll probably keep watching no matter what, so whatever. 
I just got done watching the episode “The Wedding of River Song” where in an alternate reality, created by River not killing the Doctor at Lake Silencia and thus dorking up time and killing reality, River and the Doctor did get married. And I can see them hanging out and getting so lovely dovey that Doctor 11 tells her his name that she then gives to Doctor 10 in the library episodes to make him trust her.
So I now I agree with enalzi that her story is wrapped up and we won’t see her again.
This would make an awesome spin-off series. Where we never see the Doctor except for maybe glances of his back as he goes off screen immediately while the hero is moping up the Doctors’ messes he leaves behind like in the episode Chimera mentioned. I think it would even be better if the hero wasn’t all goody-two-shoes and did some bad things to mop up some of the messes. Like in paving stone girls’ instance the hero can not find a way to make her human again so the hero decides the best course of action is to kill her.
No speculations and no spoilers, but having seen the first three leaked episodes, all I can say is I really, really, REALLY, R-E-A-L-L-Y miss RTD. 
That said I actually do like the new Doctor. It’s only the stories, arc, and companions that bother me.