While Paladin can solo, I would not say they are "great" at it. They do better against undead though of course.
Paladin soloing is mainly Pacify pulling singles, tanking and nuking (nuking for undead only of course), and rooting the mob so you can back up and heal yourself when needed. When mobs run, chain stun or root them while you finish them off.
Use Root to seperate mobs if you get adds. Keep that Lay Hands on a hotkey. You don't get Divine Aura until level 48 but once you have it it could save you from some ugly deaths if you remember to load and use it.
Potions help a lot. Damage shield, haste, heals.
At lower levels Paladin could fear kite undead, but without snare that can be too much work. Our fear doesn't last very long, and you run the risk of getting adds while you chase it around or wait for it to come back.
With your Instrument of Nife undead proc buff and your Ghoulbane you will probably want to try to focus on undead. Most of them don't run on low hp, but there are exceptions. Later you will also get a buff called Divine Purpose, which is a slow drain on your mana to add HP regen. It's basically stacking hp regen, and will help you solo too.
With a Shaman duoing becomes much better than soloing. Pacify pull to get singles if you can, Root adds otherwise. Shaman slows and sends in pet and lays down some dots. Paladin uses Stuns to maintain agro if the Shaman is casting too early or too much. Again chain stun or root the runners to finish them off.