Resources: http://www.kotsu.city.osaka.jp/foreign/english/index.html
http://www.westjr.co.jp/english/global.html

Whats the local transport like?

It mainly consists of the Municipal subway lines (9 lines) that criss-crosses the centre of Osaka city.
The JR line circles the perimeter of the city, thus called the JR Osaka Loop line.
And a couple of other railway companies travel from the major stations like Umeda and Nanba to the outskirts of Osaka City.

The Midosuji line is Osaka's equivalent to the JR Yamanote line in Tokyo. It connects Umeda area to Nanba area, which are the 2 major sightseeing and shopping locations in Osaka.

What's the best payment method?

As with the Suica in Tokyo, Osaka has the Icoca.

While the Suica can work on only the JR lines in Osaka, the Icoca works on all the subway lines, all JR lines and some other railway lines in Osaka.

The Icoca is issued by JR west, and can be bought at ticketing vending machines or the Green window office in JR stations.

Another prepaid card would be the Rainbow card, somewhat the Osaka Municipal Transport Bureau's version of the Icoca.

The difference is that the Rainbow card is not re-chargeable, and you have to feed it in the ticket slot instead of tapping.

Valid on the "Surutto Kansai network", in other words all subway lines, all JR lines, some other railway lines, tram system and local buses.

Purchasable at Rainbow card sale offices in major subway stations.

In both cards there are no discounts for travel.

And there is the Multiple ride card, purchasable from the subway ticketing offices.
For 3000yen, you can get 3300yen worth of travel value.
In other words, an extra free 300yen value.
Only problem is that most of these ticketing offices do not have english speaking staff, and only valid on subway, tram system and local buses.

What other options are there?
Depending on the extend of travel, there is the One-day Pass that costs 850yen, but only valid on the subway, tram system and local buses. But unlimited number of trips in 1 calendar day.

There is also the No-My-Car-Day Pass which costs 600yen. Similar concept as the One-day pass but only can be bought and used on every friday and 20th of the month (If it falls on a Sunday or Public Holiday, subsequent day will be No-My-Car-Day instead)


To wrap up... Osaka is easier to travel around as compared to Tokyo due to its size, but JR pass holders may have to be creative about the routes they take.
Feel free to email me for advise.
http://www.hyperdia.com works as well, so do use it for your transport planning.