If you just want to level, every class works fine. If you want to solo Heroic 4 Missions or Flashpoints you need a Tank Class w/ Healer Companion or vice versa.
In my opinion the strongest solo class is the Medicine Operative. Companion Healers dont cut it in tough fights, but you as a healer can keep a geared Kaliyo alive all day long. This combined with stealth, a great amount of CC and the ability to deal damage while your HoTs are rolling is stronger than any Healer/Tank Combo out there.
Most healers dont have the time to deal damage while keeping the companion alive in tough fights, but the Operatives healing style suits that just fine.
If you're playing a Medicine Operative and not soloing the heroic missions, you may wish to run with a DPS companion once one is available - which is at the end of Alderaan for the Agent. Things will die a lot faster, but you will need to bear in mind that your companion will be more fragile and won't be as good at keeping enemies off you.
You can certainly complete the Heroic 2 missions without much trouble as a healing-spec Operative. Just be careful to keep an eye out for roaming enemies and also be sure to use your Sleep Dart crowd control from stealth. I found the Operative to be a very good class for when things go bad. Towards the end of the Act 1 storyline, there was a boss fight where my tanking companion bugged out and ended up stuck under the floor where I couldn't heal her. I was able to complete the fight as a healer without any serious problems.
If you don't wish to play that class, all of them are viable for levelling solo without doing any heroic missions at all. Just be sure to keep yourself and your companion geared, remembering to spend your commendations on mods for orange gear, and be sure to complete all the class, planet and bonus missions.
I would say that for a new player who is getting used to the game, you may find it easier to choose a class that can heal or that gets a healing companion early on. My first character was a Sniper and I died a few times until I got used to the playstyle. Picking up my healer on Taris helped a lot as well. Great class to play, but you need to be aware that you might suffer a few deaths early on.
It's worth considering what role you would prefer to play as. If you want to play a tank or melee DPS, the Sith Juggernaut, Sith Marauder, Powertech (Bounty Hunter) and Jedi Shadow (Consular) all get their healing companions early on, which can make things a lot smoother. If you want to play a healer, the only classes that don't get a tanking companion early on are the Bounty Hunter and Trooper - and I never had problems running with a DPS companion on my Commando. (I haven't played a Mercenary-AC Bounty Hunter, so I cannot comment on that.)
Ultimately, I would advise you to play what draws your interest. It's no good me advising you to pick, say, a Sith Sorcerer if you hate playing caster/healer types.