For the wallboard joints at the inside corners, you will use a corner trowel tool; there is no need for a mud pan. Just dip the corner trowel right into the joint compound bucket and coat the corner joint well.
Step 1) Start at the top of the wall and press the compound into the corner, holding the blade at a low angle to the wall. Work your way down to about the center, then start at the bottom and work your way up. Working this way lets you avoid ending up with a big glob of compound on the floor. You want to get the compound pressed into the nailhead dimples and the gap between the adjacent drywall sheets.
Step 2) Cut a length of joint tape which will cover the entire corner joint. Slowly fold the tape in half along its length, being careful to fold it evenly. Align the center of the fold in the tape with the joint, and cautiously press your tape into the corner joint. Do not squeeze all of the joint compound out from underneath the tape, but press it well into the joint. Be sure the tape sticks to the wallboard.
Step 3) Bury the joint tape in compound by coating it as in step 1. You should not see the tape when this step is done, but at the same time, you should not have so much compound in the joint that it will crack when drying. Remove the excess at the edges with the 6 inch compound knife.
Step 4) Repeat the above steps for the joints between the walls and ceiling wallboard.
Now on to the outside corner beads>>>