In order to climb to an altitude (say 10'000 ft) you have to:
1) on the runway:
Flight Director: on
Autothrottle: on
(these are the two flip switches on the left side of the MCP)
set altitude: 10000, don't push or pull anything
manually advance throttles and release brakes (hopefully you have already configured the flaps, the autobrake in the RTO position and armed the speedbrakes
)
2) at take-off
At VR (you'll have to program the fmc in order to get that, since it depends on a lot of factors, let's say around 160 kias for simplicity's sake) gently but firmly pull the control column towards you in order to reach an attitude of approx. 15 degrees nose up (you have to get there veeeeery gently: no more than 2 degrees per second).
3) in the air
hold 170 to 180 kias up to 1'000 ft, then accelerate to 250 kias by pushing the nose down (gently).
You just have to push CMD (to select an autopilot, A or B it doesn't matter) and then LVL CHG to make it climb and stabilize at 10'000 ft.
4) climbing to cruise altitude
Same drill: select with the knob (say 27'000 ft), then depress the LVL CHG button
The trick is to understand that only selecting the altitude does nothing. The autopilot has to have command (the CMD part) and to have a go! signal (the LVL CHG part).
Note that doing so you still have to manage the throttles and the heading by yourself.
The "real thing" would be to define a flight plan in the FMC and use LNAV and VNAV, but you'll have to read a good deal of stuff before doing these things...
Best regards