If you can only afford day-trips from Osaka, I think the most south you can reasonably reach in the Kii Peninsula is Shirahama for a day excursion before returning to Osaka at night. The one-way train journey is slightly more than 2 hrs. To check train schedules, use this site:
http://www.hyperdia.com/cgi-english/hyperWeb.cgiTo see Tama the station master cat, you have to take a private train from Wakayama City to Kishi Station. JR railpass is not valid on this rail line, you have to pay separately. The one-way journey is about 30 min.
Day-trip suggestions from Osaka: Hiroshima and Miyajima, Kurashiki and Okayama, Himeji and Kobe, Nara and Horju-ji, Koya-san (reached by private Nankai rail from Osaka Namba)
If you donft mind a longer train ride (more than 2 hrs. with a train transfer), you can also consider Amanohashidate. Weather permitting, scenery there is beautiful. You can take a short ferry ride and feed the sea gulls, then take cable car up mt. for view of the famous tree-filled inlet. The interesting part is that you have to bend over and view it between your legs to apprehend its gbridge of heavenh fame. Note that to take train to Amanohashidate, you have to pay a supplement as the last leg of the journey is run by a private rail line.
Another shorter but equally interesting side-trip from Osaka is the Momofukku Ando Instant Ramen Museum which is just 20 min by train from Osaka. Though geared more towards kids, my wife and I fully enjoyed the visit there, having fun and feeling young again among all those Japanese children and making our own cup of instant noodle to take home as a souvenir. On the same rail line, 30 min from Osaka is Takarazuka, home to the famous all female Takarazuka performing group and the Tezuka Osamu Manga Museum. Again this rail line is a private rail line, you canft use JR rail pass.
The nearest subway station for Tsukiji fish market in Tokyo is Tsukiji Shijo Station on Toei Oedo Line. Slightly further away is Tsukiji Station on Tokyo Metro Hibiya Line. To see the auction, you have to be there early, around 6.00 in the morning. Try not to get in the way and no flash photography. Because it's popular with tourists, I hear nowadays the auction area is off limit and you can only watch from a distance.