Note that many exercises do not have formal names and so different sensei may call exercises different things. These are what we call them.
We always start (and end) class with Chokutotsu san bon of course 01:57 but there are a bunch of other exercises to do which keep things interesting. For all of the basic exercises there are many things that could be focussed on obviously but I tend to try and have one or two things per exercise which are my primary focus before I even start. That way I don’t go easy on my self by thinking “my zanshin was bad but the strike was good”; if this particular exercise is all about good zanshin then it doesn’t matter if my strike was good or not. The following are the things I tend to concentrate on for each exercise when I am performing the strikes.
00:10 To begin a class with we often do “Yukuri” (slow and relaxed) strikes and ashi-sabaki (footwork). Often starting doing strikes with no footwork then with sliding steps only then with fumikomi (stamping steps). When doing these type of exercises it is good to pay attention to your basics. I normally get in trouble for a lack of hanmei (side on stance) or not having my front wrist in the correct position.
00:49 To get the blood flowing mae tsuki (one or more steps followed by a strike) or tsuki nagashi (strike then run forwards a few steps) are good. My focus is usually on smooth and fast ashi-sabaki. No bouncing up and down and a strong nuki (return to kamae after the strike).
Dai Ichi Kyoushu 02:23 is a great exercise for practicing the feel of centre when you are striking. The strike sequence is omote, ura, omote, shita, nodo (or omote if practicing without men and trying to be safe). After each strike the motodachi retreats a single step and, in time with the strikers nuki, attempts to cover the mokuju. This covering opens the next target in the sequence. When done well the striker will feel the opening created and smoothly perform the next strike without large movements. Dai Ichi Geiko can be considered a form of Renzoku (continuous) tsuki. As is the basic Tsuzuite (continuous) tsuki. However the concept of Dai Ichi Geiko and Tsuzuite Tsuki can be applied to any of the other strikes and can also be performed with varying mai-ai 02:36. I have even seen Dai Ichi Geiko used as the basis of kakari (attacking) geiko.
Idoukan (Moving) strikes 02:55 are the next level of complexity in our basic drills. For this both the striker and motodachi move as if in shiai and after the strike hiki nuki (retreating) zanshin is performed. As the striker moves in and applies seme (attacking pressure) the motodachi tends to open slightly and then not move until the strike lands so as to let the striker make a clean and correct strike. The focus for me in this exercise is judging the distance quickly and performing good zanshin. I also know I am a lazy, lazy man and try to ensure my zanshin does not include a rest period; I want to be balanced and ready to go forwards immediately.
Omote no tsukaeshi 03:45 is an exercise I think of as comparable to Kiri-Kaeshi in Kendo or Uchi-Kaeshi in naginata. I tend to try and focus on speed of my strike and relaxation after my strike. The timing of striking as soon as your partners mokuju has moved out of the line of attack is very difficult for me and if I am not relaxed never results in a fast and shime (locked) strike.
Engine is an exercise where all I focus on is not falling over and not collapsing 04:02.