Which actuators are suitable for your application depends very much on what kind of robot arm you want to build. Once you have decided on what kind of arm you want you can decide on a suitable actuator for each axis.
Assuming from your description that a gantry robot wouldn't be viable, then depending on your specific application, you may want to consider a SCARA arm over an Articulated arm, which is what most people think of when they think Robot arm.
The big advantage of a SCARA arm is that most of it's lifting strength is in it's bearings. The main shoulder, elbow and wrist (yaw) joints are in a flat plane, which means that the motors only need to be strong enough to produce the lateral forces required, they don't need to support the weight of the remaining axes.
The Z axis, pitch and roll (and grip obviously) all have to work against gravity, but the Z axis is easy to gear highly enough to be able to support plenty of weight, and the pitch, roll and grip axes only have to support the payload weight, not the weight of other axes.
Compare this to an articulated arm, where many of the axes have to support the weight of all axes further down the kinematic chain.
Typically a gantry robot will use linear actuators for the main X, Y & Z axes. These could be low performance, low accuracy, high force actuators such as a lead screw with a servo or stepper drive (force and performance can be traded but accuracy will always be limited by backlash), all the way up to high performance, high accuracy direct drive linear motors with precision encoders.
The remaining 3DOF manipulator will usually require precision rotational motion for pitch, roll and yaw, so usually an electric motor (either stepper or servo), will be most suitable. Even a small motor with a reasonably high gearing can resist gravity against quite high loads.
An aside on servo motors vs. stepper motors
The difference between servo(1) and stepper is a trade-off between complexity and certainty in control.
A servo motor requires an encoder for position feedback, whereas a stepper doesn't. This means that a stepper is electrically much simpler, and from a control point of view simpler if you want low performance.
If you want to get the most out of your motor though (pushing it close to it's limit), then steppers get much more difficult to control predictably. With position feedback on a servo you can tune performance much more aggressively and since you know if it fails to reach it's target position or velocity then your servo loop will get to find out about it and correct it.
With a stepper you have to tune the system so that you can guarantee that it can always make the step, irrespective of the desired speed of move or weight of the payload. Note that some people will suggest adding an encoder to detect missed steps on a stepper motor, but if you are going to do that then you might as well have used a servo motor in the first place!
With a SCARA arm, the Z axis is probably the only linear axis, while the remaining axes can all be done with rotational motor, so again stepper or servo motor. Sizing these motors is relatively easy because the weight carried is less important to many of them. The motor required to overcome the inertia of a load is rather less than sizing it to overcome gravity.
With an articulated arm the calculations are more tricky, because most axes will need actuators sized depending on both moving the load and lifting it, but again an electric motor is the easiest to control and use.
Finally there is the gripper. This is where I have seen the most variety in actuators. Depending on your applications you could easily use any number of different actuators.
I've used systems with traditional motor driven grippers, linear actuated grippers, piezo flexture grips, pneumatically actuated grippers, vacuum pick-ups and simple slots or hooks amongst others, many of which were specific to the application. What your typical payload is could significantly change the actuator which is best for you. (2)
Doing your calcs
As Rocketmagnet suggests ultimately you are going to have to break out your calculator.
You will need to take into account kinematics of your system, the maximum load on each motor (taking into account the worst case with arm fully extended if you are using an articulate arm design), the speed (a smaller motor with higher gearing might give the force you need without the speed, but a beefier motor might give you a higher torque with lower gearing and a higher speed etc.) and the positional accuracy you need.
In general, the more money you throw at the problem, the better performance (speed, accuracy, power consumption) you will get. But analysing specifications and making smart purchasing decisions can help optimise the price/performance of your robot.
(1) Note that my experience is with industrial servos, typically a brushed or brushless DC motors with a rotary encoder, so this may or may not apply with hobby RC servos.
(2) I would suggest posting another question on this.