Army, Navy and RAF always have caps. Use Royal Navy only when necessary to avoid confusion with the merchant navy (lower case), otherwise Navy is fine. RAF is not spelled out unless there is some reason to do so.
military ranks: In compounds such as Major General and Lieutenant Colonel, where two ranks are joined, give the full rank at the first reference then the more senior one at the second, eg Lieutenant Commander Christopher Roberts, then Commander Roberts or ‘the commander’ (lower case). If the officer has a knighthood, that takes precedence in subsequent mentions, eg General Sir Clifford Martin becomes Sir Clifford, or ‘the general’ (l/c) for variation. Rear Admiral and Vice Admiral are shortened to Admiral. If possible, do not abbreviate except in lists. These compound ranks are not hyphenated.
Most two-word ranks which are not compound are not shortened at subsequent mentions, eg Second Lieutenant, Lance Corporal, and Air Commodore. It is a lot better not to abbreviate these ranks unless you are really pushed for space. Warrant Officers may be Class 1 or 2 (not 1st or 2nd Class), and you should state which, but subsequently all can be called ‘Warrant Officer’. If you want to abbreviate, used ‘WO1’ or WO2’.
police ranks: Spell out in full at the first mention. Subsequently officers of the rank of Chief Inspector and above should be referred to as Mr, Mrs or Miss. Inspector and below should continue to be used at subsequent mentions, preferably not abbreviated. Use PC, Pc or ‘the constable’ in headlines and copy. Strictly speaking, there are no WPCs now (they are all PCs) but WPC is often used in headlines as a shorthand.
Private: Many regiments have their own name for Private, the lowest rank. For example a Private in the Royal Engineers is a Sapper, in the Royal Artillery a Gunner, in the Duke of Lancaster's Regiment a Kingsman and in the Royal Regiment of Fusiliers, a Fusilier.
aircraftman, aircraftwoman: RAF ranks. Note that there is no ‘s’ in the middle.
court martial, plural courts martial (l/c): Findings (note: not verdicts) and sentences are subject to confirmation and this should be mentioned. A Naval officer may be dismissed his ship, not dismissed from his ship.
the Last Post and Reveille are sounded, not played.
Welch Regiment, Royal Welch Fusiliers: Now both amalgamated with other regiments but you may need the correct spelling.