Summary
AOTC gives some information about the size of the Republic. It contains
on the order of tens of thousands of systems

in
a reasonably well explored galaxy, although a few areas are off the
charts. The Republic thinks very highly of its own geographic skills,
although this may not be entirely justified.
AOTC is unique among all Star Wars films in providing two visual maps
of the galaxy.

The maps
do clearly establish the Star Wars galaxy to be a spiral galaxy,
although its size is not clear.
The Details

Count Dooku
anticipates that
"ten
thousand more systems"
will
rally to his cause, splitting the Republic
"in two."

Jocasta Nu
states that if something isn't in the records, it
"doesn't exist." This is
demonstrably incorrect in the case of Kamino. It may be argued that
this means that the Republic has thoroughly mapped the entire galaxy;
at a minimum, we are to accept that inhabited planets within the
Republic's neighborhood should all be on the charts.

In the left
map below, seen on Padme's yacht, the boxed area is zoomed
in on to provide an immediate view of how close Tattooine and Geonosis
are to one another. These are Outer Rim systems
"halfway across the galaxy"
from
Coruscant,
"less than a parsec"
apart. The map to the right, seen in the Jedi Temple, zoomed in on
Kamino.