You won't be doing much as far as skills go. All of those classes have 2+INT per level. Your CHA will be good, so you can invest in Diplomacy and what not... but thats about it.
This is an interesting thought exercise. Hrm.
The game will be starting at lvl 4, Core Rulebook only, 25 point buy. Only core races, leaning towards Human or Half-Orc, though I can be persuaded otherwise. No traits, literally only the core book is an option.
I would definitely go with Human over Half-Orc, simply for the bonus skill rank per level since you've said you'll need skills. The feat doesn't hurt, either.
25 point buy helps, though lack of options beyond Core is definitely making it a bit difficult to decide which way to go. I'm stuck between these two:
Dexterity 12
Constitution 14
Intelligence 12
Wisdom 8
Charisma 14
With this one, you'll need to prioritize getting a +CHA headband early enough to allow you to get past 4th level spells. With the multiclassing, this might be somewhere around level 12, so easily doable.
Dexterity 12
Constitution 14
Intelligence 12
Wisdom 9
Charisma 16
This is more balanced between Str/Cha. I tend to like it a bit more. You won't be quite as strong when attacking outside of smiting evil, but on the other hand, your saves will be a bit better and if you get hit with a wisdom drain, you won't be quite as weak there either(only one point difference, but I had to put the leftover point somewhere). You could also drop 2 points from Dex and one from Wis to up Str by one more, giving you another point of modifier come level 8.
Regarding both: I normally don't like dumping Wisdom, but starting a paladin high enough to add Cha to saves, it's reasonably acceptable. With 2+int+racial for skills, that'll be 4 per level. Not as high as I like, but it is what it is.
With a focus on buff spells, you don't have to worry about shoring up your Charisma quite as much to deal with spell DCs, but it's still important for the increase to saves as well as other paladin benefits. But since you won't be full classing paladin, the reduced number of smites per day also limit how much it helps you as well.
Suggestions for building the classes
There are a few options here. As I see it, start with 2 levels in paladin, regardless of what else you do. From there, I see two other routes I would consider.
The first: 1-2 Levels in Sorcerer, rest goes paladin and DD when you can get it. To me, anything beyond is a painful loss of BAB on a build that already kinda feels the loss of it. Still, your breath weapon will benefit from the higher bloodline level.
The second: 2 levels paladin, 3 levels in Bard, DD to the point where you're happy with it, then rest paladin. Gives you three levels of increased skills, Versatile Performance further streamlines two of your skills into a single one, and Inspire Competence helps your party make their skill checks when you can't, and now you can also make all knowledge checks untrained.
The bard spell list isn't as good as the Sorc/Wiz one, but it has some decent buffs. You'll feel the pinch from lower bloodline, but it was never gonna be ultra high with either method I'm suggesting anyway.
I'd choose the bard one, personally, but it's up to you ultimately. And I'm not entirely sure either option is the best overall, but hopefully at least they help a bit.
You are asking for a lot out of a limited palette. I think you have to go bard and fluff the dragon aspects until dragon disciple at 6th level. Otherwise, you are a paladin/sorcerer with either no armor or spell failure woes. Bard isn't a bad gig to fluff. Be a lightly armored knight with sword and shield. Lots of skills and knowledge are racial memories of your draconic heritage. performance abilities are moments of dragon awe showing through. Choose spells that fit the concept and can be made dragon-ish.
This is the way I'd go points wise.
Str 16, Dex 12, Con 16, Int 10, Wis 10, Cha 15
You don't need a 20 stat at first level. And if you're really looking at a big weapon, I'd think bastard sword because it gives you options of leaving a hand open for shield or casting use.
That's either a 29 point array before racial modifiers or a 24 point array after.
And you can take one hand of your greatsword to cast, as a free action.
That looks to get me BAB 16, CL 12, 11 fort, 6 refl, 13 will. Good HP, and it progresses nicely, without too many dead spots where I am weaker than need be. Thoughts?
I like the EK for gaining both caster levels and good BAB, nice touch.
Are the saves before any magic belts/headbands?
Those are base saves before paladin stuff, stats, or items.
Nice. Those saves, 6th(?) level casting and a pretty decent BAB should make you an excellent switch hitter.
It's just too bad that there's nothing beyond Core you can use, since there are some things in the APG and Ultimate Equipment which might enhance your abilities even further.
To me the biggest thing you're missing is Eldritch Heritage so you can toss in another bloodline and get the inherent strength bonuses (Orc, Abyssal).
I've done this character myself in a way. What level are you going to? You'll start off struggling, but it pays off late game very well. Around level 14 I was unstoppable (and this was with claws only, so a claw build IS viable).
I have settled on Human, and im considering Paladin 4, Sorcerer 2, DD 8, EK 5 at the moment.
That's a good mix. I've never done Eldritch Knight, and I went with much more Sorcerer for better casting, but I'm willing to bet that this works great.
BTW, if skills are important, I think 12 in intelligence might be a pretty decent idea. Paladins have a pretty decent skill set, pity they don´t have an archetype for more skill points a la tactician fighters.
By the way, I have 4500 gold to play with. Thoughts on gear? We will be dungeon crawling andIin the wilderness where skills are much more valuable alot, so many skill many of the skill items and neat tools like Rope of Climbing have caught my eye.
If your GM is big on nighttime attacks, you might consider a Ring of Sustenance, though that's more than half your gold, so might want to wait.
Another thought is to go with a couple of the big Six(+1 to weapon and armor, if it were me), then spend the rest on consumables and more mundane tools.
Check out this guide for useful cheap magic items and mundane tools that you might want to consider in any character's arsenal. At the very least, I would devote a few hundred gold to purely mundane items.
Discuss with your GM what types of masterwork tools you can get. They're rather cheap for a +2 circumstance bonus on a related skill(might be a situational use of the skill, like a masterwork snare kit that allows +2 on survival to find food). Ultimately it's up to him what will fly, so you'll have to sit down on those. With that kind of gold to start with, I'd shoot for at least four such items in the party, personally.