Administer Warning and Technical
 

I looked in the Official's Manual but maybe I didn't look hard enough for this question. If I ever need to get a warning recorded (for a coach, or if defender's arm crosses OOB line on an inbound play) what exactly should I say and any mechanics for this? Also what do I say and what are the mechanics for recording a bench technical or player technical? I know to make the "T" mechanic but anything else? I'm not anxious to give anyone a T, but I wanted to know to make sure I know how to handle it if it ever does happen.

Depends on how much you trust your table. ;)

For the T, report it as you would for any other foul. "Blue, 45, technical foul". If you want, check with the book to be sure they also record it as one of #45s five allowable fouls, and add it to the team total.

For the warning, "Warning on White for xxxxxxx" If you feel the need, add "please record a ..." in front of that. No need to get all the way to the reporting area for this (imo -- but I haven't really read the manual on this part). It might prove helpful if whichever official is nearest the table (if not the calling official) asks them to record it.

If a defender reaches through the boundary plane for the first time, you have a delay of game warning. Make sure the scorer notes it in the scorebook and make sure you inform the head coach that its his warning also. Subsequent violations result in a tech.

Similar but Different...Sorta
 

If I recall correctly (and others will respond soon to correct me if I'm wrong), you've cited two different things: warning for a coach (optional courtesy we seem to extend prior to a whack) and team warning for delay of game (4-47) which is then recorded in the scorebook.
Probably best not to consider the two similar.

NFHS Officials Guidebook explains this in 2.5.3.

So I go to the table and have a warning recorded for delay of game but if coach needs a warning that is just verbal?

Correct. From the "Officials Manual":

B. Warnings for Behavior:
1. Though no provision provides for this in the rules book, many officials
issue warnings to coaches and players for behavior that is not egregious
(not warranting a technical foul), yet is inappropriate and must
be addressed.
2. Because there is no provision in the rules book for a behavioral warning,
officials should obtain guidance from their state high school association
regarding this issue.
3. Warnings must be issued professionally and in a conversational tone
with the offender. Raising your voice further antagonizes the situation
and the person you are trying to address. The official must be the voice
of reason and keep the situation under control.
4. A warning of this nature is typically given verbally (“This is your warning
coach”) and visually (holding up the one-handed “stop sign”).
5. Only one warning should be given to an individual, per game.
Therefore, communicate with your partners at the first appropriate
opportunity that a warning has been issued. If the behavior continues,
issue the appropriate penalty immediately.
6. These types of warnings are not documented in the scorebook.

Your state may have different directives. Our state has its "Officials Guidebook" available online. Check with whatever state resources you might have available to you.

Some areas / officials like to have the coach warning also recorded.

Coaches do seem to "take it more seriously" when it is recorded. I agree it shouldn't matter and the officials need to take care of business regardless of whether it is recorded.

I also inform my partner(s) of any warning, correct?

It should be obvious to them, but yes.

Correct.

BTW, one point of controversy in this "Issue a Warning before Whacking a Coach" thing is this: Say you issue a warning to Coach A, putting him on notice that one more episode and he's sitting down. Then, thinking that he'd only get a warning, Coach B goes off on you but more vehemently than the first coach did and you decide to T him up immediately. He protests that "Wait a minute, you gave that guy a warning, why aren't you giving me a warning first? Blah, blah, blah."
See the can of worms it opens up?
Anyway, that's been one line of argumentation against this whole "issue a warning first to the coach" thing.