A bard without spells isn't a bard at all. You most likely want the Ninja or just a charismatic rogue if you're going down that route. There are variants of other classes that get the bard's Inspire Courage ability in other books, but I'm afraid the spells are at the very heart of the class and cannot be removed without starting from scratch.
The Paladin is among the most powerful healers in the game, so removing his spells will not even begin to address the problem. There is a spell-less paladin already in the Advance Players' Guide, but it has plenty of healing.
The Cavalier class, also from the APG, is a mounted knight with a code of conduct and abilities relating to his banner and knightly order. It will fit your world better.
The Ranger has spell-less variants available already which will fit your world very well. The Skirmisher from the Advanced Player's Guide and the Trapper from Ultimate Magic.
Lastly, do not make the mistake of lumping all the divine casters under the clumsy title of "healers". They're absolutely not that.
Only 50% of clerics have any aptitude at healing beyond having the spells on their list and not all of them even prepare those. Their combat role is usually a combination of buff spells, attack spells and hitting things with big weapons.
Druids have little ability to heal. They are battlefield controllers, summoners, warriors and clever shapeshifters. You could save this whole class just by removing their cure and restoration spells in favour of spells from one of the druid-accessible domains.
EDIT: I know you wanted to stick to core, but there's no point in reinventing the wheel. Copy paste the stuff I recommended into your world book if you have to.
p6 codex! check it out
Cool, E6 is a great way to accomplish this sort of feel. I am playing in an e6 game and enjoy it immensly. Campaign is more then a year old and the party members are 'world wide heroes', but a hydra, or a couple ogres is still something we approach with preparation and caution.So I am working on my world for Pathfinder, and decided to go with E6 while creating it to keep the epic and low magic feel that I love from the early levels of a campaign.
I decided I wanted a game that could be played with the core book and a supplemental guide for the E6 nature of the world alone, and also wanted one where the danger's never stopped being dangerous.
Classes:
The classes I decided to go with are Fighter, Rogue and Sorcerer. I left out healers as healing magic is supposed to be exceptionally rare, though I do include an blood line of sorcerers that can take some cleric healing and restoration spells if they like, that blood line being elves (with healing / restorative spells generally being referred to as elf-magic).
You are going to have to make some very serious changes to the structure of the game if you make restorative magic rare. Multiple encounters a day or heck multiple encounters per week becomes impossible. Natural healing on its own is really limited, and players wont be able to recover what they need to keep going past one or two encounters. While you dont have to have a dedicated healer in a party, access to hit point recovery and restorative spells (even in e6 things like lesser restoration are really important), fundamental issues in the game are going to be popping up all over the place.
One option, that will help with hit points, is to adopt an alternate set of hit point rules. Evil lincoln has a thread running around somewhere on the boards, but basically, any damage that isnt a critical hit, or the hit that drops you bellow 0 health is just 'stamina damage'. Those injuries heal with a minutes rest (recovering your breath) only critical hits, or the blow that puts you down is an actual 'wound' that has to be healed. This might allow your party to function with limited healing. But condition removal, especially lesser restore is really important if you are dealing with anything that gives out such conditions and you dont want the party to have to stop for days/weeks to recover from an encounter.
I would like to include bards, rangers and paladins some how, but I would need to rework them to be spell-less variants as magic is supposed to be rare and unknown.
Rogue genius games has a product line of archetypes that are different from normal archetypes. They all are set packages that each class can trade a set package for. For bards, rangers, paladins and inquisitors, they all have the option to trade out their spells as part of this. Theres a wide variety of things you can get in exchange, you probably would want to stick to the archer and martial archetypes products given your desire to reduce the presense of magic.
For NPC's I decided to have the classes of Aristocrat, Commoner, Warrior and Expert, with most NPC's falling into these categories. I left out Adept as I felt that magic, while rare, should be powerful and frightening when encountered, and sorcerers can already fill that role.
Advancement past 6th level:
After 6th level the players have access to epic feats. To keep things interesting I made it so that sorcerers can gain access to a single fourth level spell slot and spell and the other three classes can gain access to a single 'prestige class' line of feats, such as assassins for rogues, which gives them access to some of that prestige classes abilities. Each class would ideally have several prestige classes to choose from, path wise. Sorcerer's bloodlines also have more 'blood line' related level 6+ feats that would add flavor and individuality to the bloodlines.
E6 already has rules for this you might want to check it out. Basically you can get a limited set of 4th level spells through epic feats.
I also decided to go with sorcerer over wizard for two reasons. The first was balance, in that by limiting them to a small selection of spells it keeps a spell caster from eventually acquiring basically every spell and being much more useful in many situations than other characters. It also makes most spell casters unique as very few will have the same line up of spells.
Not a bad method, though in a low magic setting the nature of vancian magic seems kind of counter intuitive. If magic is rare and mysterious, the complex and precise nature of core spells is a bit off. If you want a more elemental and 'primal' kind of magic, that is less of a utility belt, you might consider using the Riven Mage. They dont use standard spells, and instead have a small selection of basic spells that they can put more or less of their energy (called flux) into for greater or lesser effect. It also has very limited acces to healing (with a minimum level of 5th and a requirement to take a certain specialization) which might suite your theme rather well.
So I guess my questions would be does anyone who has run E-6 have any advice, and anyone in general have suggestions for making bards, paladins and rangers fun without spells while keeping them distinctly different from fighters?
As I mentioned, use the archetypes to trade out spells. And also consider carefully how you are going to handle hit points and healing magic if you dont want to pace of your game to bog down.
I second Helaman's Strain/Injury suggestion. I use a pared down version in my own games:
Strain and Injury
Most damage doesn’t represent actual injuries. Such damage is called 'strain'. All strain is removed after a few minutes rest and refit in relative safety and comfort.
Damage from failed saves, critical hits and the final blow which reduces a creature below 0 hit points is 'injury'. Injury never goes away without treatment.
Healing magic always removes injury before strain.
Non-lethal damage is not tracked separately. Instead, damage from non-lethal sources is always strain.