Which is the best race/class combination and tradeoffs involved for solo play in Lord of the Rings Online without the need to to be in a group most of the time?
The best class to solo with in LOTRO would be the Rune-keeper. The Rune-keeper can swap between healing and offensive spells, so you can pretty much deal lots of damage, and if you ever find yourself dying, stop dealing damage and heal yourself. (Note, with the rune-keepers' spell system, you need to build up healing or offensive energy - so if you've been casting offensive spells for a long time, you'll need to stop and start casting the low-power healing spells to build up healing energy, before you can heal yourself for any great amount of HP).
The class I've been soloing with is the burglar. He is good at soloing for the reason that he can disspapear if things get messy, and he can handle a large group of enemies better than other classes, with about 4 stuns and various debuffs.
Other than that, the Guardian - with a tank spec - can solo relatively easy, though it will take you longer as you won't be dealing a lot of damage.
Lastly, The Hunter is a pretty good soloer, as he can lay down traps and deal damage (more than the burglar can) from a distance, and can teleport away from combat if things get hairy.
Unless you have a class preference, I'd say go with the Rune-keeper or the hunter.
I've played a bit of every class -- in fact, managed to get all but Runekeeper to 20 without dying.
There are several considerations when soloing: How long it takes to kill things, how much recovery time between fights, and 'disaster mitigation' survivability. For soloing purposes, I personally prefer to max out on the latter two.
Lore-master is, in my mind, a good soloing option for an advanced player. You have pets, crowd control, and heals. Grab the trait to remove the cast time on your Blinding Flash, and you're well suited to handling multiple mobs at once. You can either go for a tanking pet (The bear) or a quick-attacking pet to maximize your flank heal. Note that building for DPS will negate most of your mez, if you choose that route.
Tanks like the Warden and Guardian are excellent as well. You won't rock the DPS but your survivability is high. A Warden, in particular, has enough self heals that by higher levels you'll prefer larger pulls, as your AOEs will damage them all (and heal you that damage). The Warden also gets the same teleporting ability as the Hunter, although only for themselves rather than a party. (Which is fine for soloing)
The Runekeeper is a glass cannon. Excellent for soloing at lower levels, he runs into more trouble in the higher ones. You need to be much more careful on your pulls than the tanks or Loremaster do. Against a single opponent, they're dead before they reach you; against a group, you'll be dead where a Warden could simply tank them, and the LM would have rooted the melee, mezzed the caster, sent their raven to distract the archer and beat the last one to death with their staff.
If you choose to level through skirmishes, I recommend an herbalist for the Warden/Guardian and one of the DPS for the Loremaster -- but given your pets, you can't really make a bad choice with the loremaster. (I personally use an archer without any DOTS, so I can keep my targets clean in case I need to mez them.)
Regarding races -- it really doesn't matter. None of the bonuses are that stunning; I might recommend an Elven Loremaster for the ability to occasionally teleport to Rivendale, but you're probably just as well to invest in a second 'home' map location.
The Warden is an excellent class to play solo as well, as she's a tank that is able to deal some damage nonetheless. Plus, she can teleport to the same locations as hunters. (No campfire return or fellowship taxi, though.)
In combat, Wardens combine three basic attack skills (spear, shield, taunt) in chains that are up to 5 skills long and give you a special attack (gambit). This is not as complicated as it seems at first, since you won't be using all possible gambits anyway. Later on you will also be able to learn two-skill "macros" that speed up gambit building.
The Warden wears medium armor (which may seem a bit peculiar for a tank), but has some self-healing gambits.
While Rune-Keepers (RK's) are nice because they can both heal/damage (although not at the same time), Hunters are nicer for soloing because of the ports. Starting at level 20, Hunters will aquire a series of skills that let them teleport to nearly every location in Middle-Earth (ME). They also have a skill called Return to Camp. This lets Hunters teleport to Campfires that are set in various locations (click here for list) across ME. However, all the ports require Travelling Rations. These are not hard to come by, you can buy them at a Provisioner for 26silver/100rations.
Another nice thing is that Hunters can also bring their entire fellowship with them when they port. However, the members must all be near the Hunter when this happens.
For the damage aspects, Hunters are better than RK's because RK's require time to build up their most powerful skill. In order for RK's to use their most powerful skills, they must first use weaker skills to build up Attunement. Once their Attunment reaches a certain level, they can use more powerful skills. However, Hunters can use their most powerful skills whenever they want (assuming they have Power for it), excluding two: Merciful Shot and Agile Rejoinder. Merciful Shot can only be used once the enemy has fallen to less than Half of its health. Agile Rejoinder can only be used when you Parry an attack.
While RK's use Attunement, Hunters use Focus. A Hunter can instantly gain Full Focus (9 points) by using a skill called Intent Concentration (in and out of combat), or they can do it over a few seconds with a skill called Focus. However, the skill Focus can only be used Out of Combat. Hunters can also aquire a trait called "Enduring Precision" that gives them one focus every eight seconds while in Precision Stance.
The Hunter's most efficient skill is Penetrating Shot. This is aquired at level 2, and requires three Focus points. Three focus can either be built up by using skills like Quick Shot (gives 1 point) Swift Bow (gives two points) Barbed Arrow (gives one point), or by using skill Focus/Intent Concentration. Penetrating Shot is most used in a Focus Build.
(Now that I'm reading this myself, I'm wondering why I wrote that last part)