Overall thoughts on Sorcerer:
I think it has definitely become much more of a "finesse" class than it was previously. The apparent key to their power is the efficient use of sorcery points and metamagic. It wasn't brought up on this thread, but due to the way spells scale there are ways to do more overall damage (significantly more over the course of the day) by converting higher level spell slots into lower level spell slots and/or sorcery points to use for metamagic. You really have to study your class features and spells to figure out how best to make use of that. It isn't
hard per se, but for those of us who don't have an instant internal calculator, we need to take a bit and think through the possibilities and determine some general conversion strategies and techniques that our character is going to use. While that is focused on damage, there are also metamagics that work well for non-damaging things, like Extend Spell that not only doubles range, but allows you to turn touch into 30' range; and Twinned Spell that allows any single target spell to target 2 targets.
It's also worth noting that even if you ignore metamagic and use sorcery points only to create more spell slots, you get more spell slots out of it than a wizard's arcane recovery (remember--arcane recovery takes place during a short rest but only
once per day; a lot of people have missed that for some reason).
As far as subclasses, the draconic bloodline is a pretty strong. In addition to some draconic social skills, you get an extra hp per level and what amounts to permanent mage armor right from the get go. If you are focused on damage, you are adding your charisma mod to your chosen element from 6th level on, in addition to other features. Wild mage seems rather weak to me in the beginning by contrast, but at 14th level wild mage gets a major power boost if the DM is frequently allowing you to roll for wild surges. Rolling twice and picking which result to use on a table that has slightly more positive results than negative results is great. I did some test rolling and tried doing that (roll twice, take the best) dozens of times and only ended up having to take a bad result 2 or 3 times.
Note also that if you are willing to spend a feat you can get wizard style ritual casting (no preparation required) for the spell list of your choice. They could choose wizard and have the same ability that the wizard does, or pick a different class and expand their repertoire that way. That isn't sorcerer specific, I just thought it was worth mentioning as it's a pretty awesome feat (which would be a lot more awesome if there were about 3 times as many rituals as there are).
Maybe I'm just completely misreading multiclassing rules, but it seems to me that if you take 1 level of wizard and the rest sorcerer, you can learn a ton of spells (though you can only prepare a few from your spellbook).
The rules on spellbooks say you can only scribe spells that you have spellslots for.
The rules for multiclassed spellcasters say you add all the class levels together and look at that chart.
So a wizard 1/sorcerer 6 gets 4th level spell slots, right? Would he be able to scribe 4th level spells into a spellbook? It seems like that is sorta mitigated by him only being able to prepare a number of wizard spells per day equal to 1 + his Int mod.
Yes, you are completely misreading multi-classing rules. (Okay, actually just slightly misreading

.)
The point you are missing is that you have to keep your spells known and highest level spells castable separate for your classes. A spell you scribe from a scroll is added only to your
wizard spell list, not your sorcerer list. And since you only have access to 1st level wizard spells, you can't learn (from scrolls or otherwise) or cast any spells above 1st level from the wizard class. So you can't scribe, prepare, or cast
ice storm as a wizard spell. If you want
ice storm, you have to learn it as one of your limited sorcerer spells. What your
are allowed to do is use higher level spell
slots to cast lower level spells, regardless of what class those spells come from. So if you know
magic missile because you scribed it into your wizard spell book you
can cast it using a 4th level spell slot for greater effect.
The real key here is that spell
slots and spell
levels are handled very differently. Multiclassing
never gives you access to spells that are of a higher level in a class than you could learn from your levels in that class. 1st level wizards can't learn any wizard spells that aren't first level. Therefore multiclassed 1st level wizards can't learn any wizard spells that aren't first level, even if they have 19 levels of sorcerer and a spell is on both lists. They'd have to learn and cast higher level spells as sorcerer spells.
It can be a bit confusing if it's new, but once you get it it makes sense.
Forced specialization for Sorcs would be fine, but they have a more restricted spell list than Wizards (so you can actually only specialize in some things), and what do they get in return for having a much more limited scope (and no freebie rituals)? They cannot get more slots anymore (like 3e did it), because all casters have a unified progression - so they try to sidestep the issue with power points, but then Wizards get arcane recovery, while Sorc has to choose either slots OR metamagics...
Sorcs certainly look much weaker than Wizards. It's like the power points need to be doubled.
If they just go with spell slots they come out better than arcane recovery, but as you said, they have to divide their points between spell slots and metamagic.
I haven't seen them in action yet, but I agree that more sorcery points are probably warranted for the class.
Sorcerers don't have a narrow spell list; it's bigger than the warlock list and is roughly on par with the size of the bard, cleric and druid lists. No, it's wizards who get an expanded spell list, because versatility is part of their schtick.
The thing that bugs me is that the sorcerer gets so few spells known overall, fewer even than the bard. That sucks.
Not only is it bigger than the warlock list (by far) it is the second largest spell list. The relative size of the spell lists is:
Wizard > Sorcerer > Bard (yep, you read that right)> Druid > Cleric > Warlock > Paladin* > Ranger.
*Tied with ranger for base spell lists, but get extra spells from Oath.
Sorcerer's list is noticeably bigger than bard, and has a few spells (generally about 1-2 per level, but none at some levels) that aren't on the wizard spell list. One notable example is Enhance Attribute, the 5e equivalent of all the 3e
bull's strength, cat's grace line of spells.
I agree that known spells can probably stand to be increased. I'm not sure of the logic behind giving bards more than sorcerer.
It is possible that in an attempt to make sure sorcerer isn't outshining wizard they underpowered it, but I'll have to see it in action for a while to make a real judgement on it.
It's also worth noticing that if someone just wants to go with a relatively simple to play blastery caster, a draconic bloodline sorcerer with the right metamagic options is probably at least on par with an evoker, all things being considered.