I started my sightseeing at the Shitennoji Temple, which was founded in 593 A.D. and thus is one of the oldest Buddhist temples in Japan. None of the buildings at the temple are originals but I'd still recommend a trip here.
The other notable sights in the area are Tennoji Park, Festival Gate, and Tsutenkaku Tower. Tennoji Park contains a decent zoo(admission 500 yen) and an excellent, circular Japanese-style garden called Keitaku-en. I'd highly recommend the latter, which has an entrance directly opposite the north entrance to Tennoji Station. Also in the park are an Art Museum and a European-style garden with fountains and flower sculptures.
Nippombashi Den Den Town, Osaka's answer to Tokyo's Akihabara district, is a shopping street of over 300 stores specializing in cut-price electronic appliances. Take the subway to Ebisucho station (on the Sukaisuji line). Exits 1A and 1B will drop you slap bang in the middle of the area.
Osaka Castle is another reproduction well worth seeing.
Quick Tips:
I'd definitely recommend picking up copies of 'Subway Lines In Osaka', the fold-out 'Osaka City Map', and the independently produced 'Kansai Hideaways'. These are available at the tourist information booths at JR Tennoji Station, JR Osaka Station, JR Namba Station, and JR Shin-Osaka station. The latter is written by expats and covers Kyoto, Nara, and Kobe in addition to Osaka, while the two former publications are of great value in navigating the city.
Two good pubs I found were Tin's Hall (next to Tennoji Station, check out their web site here for directions) and the Pig & Whistle, a British pub with bars in Umeda, which is next to Higashi Umeda station and adjacent to the Sinsaibashi subway station.
Best Way To Get Around:
Transport links from Kansai Airport are rapid and extremely efficient. It took 45 minutes and cost 1030 yen on the Rapid JR train from the airport to the Tennoji area where I was staying. Most journeys inside the city itself can be done via a combination of walking and subway journeys.
The subway itself is an excellent way of getting around, with a one day ticket costing 850 yen. The 'Subway Lines in Osaka' leaflet, which is available at all tourist information centers, explains how to purchase tickets and details the location of the button offering English-language guidance. The machines are also useful for getting large denomination notes changed as some give change of 10000 yen.