Here is a summary of all I have found, including comments and
suggestions from the newsgroups:

Royal Manticoran Navy:

The normal uniform is "Black and Gold" (primarily black, though. There
is little reference to which bits are gold.

The Captain wears a white beret, all other crew wear a black beret.

There is a tunic (primarily black) (NO buttons) with epaulets, and a
turtleneck white blouse worn underneath. Trousers are black, with a gold
stripe along the outer seams (This may be the dress uniform only)

Previous hyper-capable commands are indicated by a series of gold stars
embroidered on the breast of the tunic (left?)

The is a Shoulder Flash on the right shoulder of the tunic, showing a
rampant manticore in black on a red background.

On the left shoulder is a horseshoe shaped patch with the ships name on
it.

Assumably there are also unit/speciality patches as well, but they are
not mentioned specifically in the books.

The full Mess Dress uniform includes a ceremonial sword, and there is
mention of considerable amounts of braid (for senior ranks)

Known Rank insignia:

Ensign    - 1 gold band on sleeve,
Lt          - 2 gold bands on sleeve,
Lt-Cmdr   - 2 1/2 Gold bands on sleeve,
Commander - Three Gold Bands on Sleeve,
Captain JG  - Four Gold Bands on sleeve, 4 "pips" on collar
Captain of the List - Four Gold Bands on sleeve, single "planet" on
collar
Admiral - Four broad gold Bands on sleeve, 3 x 9-pointed stars on collar

Staff offices wear a gold "aguilette" (whatever that is) dangling from
the left(?) shoulder.

Notes:
A half band is a very thin band
Pips are diamond shapes used by Commonwealth Forces (British, Canadian,
Australian
etc).

offog said:
    I want to say that RMN commodore uniforms have the same 4 stripes as
a captain of the list, but the 4th one is broader.
(not sure from where this reference come)

This is the sum total of my knowledge from scouring through the books.
The rank insignia seem to be more or less in line with the current US
Navy rank insignia, although there does not appear the be an equivalent
of "Captain (Junior Grade)" in either the US or British navies.

That being the case, the following insignia could apply:
Ensign = 1 'closed' pip
Lt. JG  = 1 'closed' pip and 1 open 'pip'
Lt. SG  = 2 'closed' pips
Lt. Cmdr. = 2 'closed' pips and 1 open 'pip'
Cmdr.  = 3 'closed' pips
Capt. JG  = 4 'closed' pips
Capt. SG  = Full planet
Commodore = 2 planets
Rear Admiral = 1 star (nine pointed)
Vice Admiral  = 2 stars (nine pointed)
Admiral = 3 stars (nine pointed)
Fleet Admiral = 4 stars (nine pointed)

(What the difference between a 'closed' and an 'open' pip I have no
idea.)
As for noncoms, nothing specific is mentioned in the books but:

"US Army uses long diagonal stripes on left fore-arm for enlistments,
short horizontal bars for combat tours (hershey bars)."

"Officers wear enlistment stripes only if earned for service prior to
receiving a commission."

Other Services:

Manticoran Marines wear green and black
Grayson Navy wears blue.
"...The short waisted tunic was a lighter blue than the trousers, which
seemed an equally unnatural reversal of the way things ought to be, and the
gold leaves on her visor...."
"...high peaked, visored cap..." (from Flag in Exile)
(and mentions archaic buttons)

Grayson Marines wear green and brown
Masadan Navy wears Scarlet and Gold
People's Navy wears Green and Gray - and the symbol for security is an hourglass and sword.