Your local line is the Keio Line, which takes you to Shinjuku in about 30-50 minutes, depending on which train or combination of trains you take. For more detailed information, click here.
Shinjuku is perhaps the busiest area of Tokyo, and to be within about 30 minutes of it is considered to be quite central. On the other hand Minami-Ohsawa only began to develop when the new campus of TMU opened in 1991, so the area does not have any traditional atmosphere. Traditional areas within easy reach are Hachioji, Machida, and Chofu.
In the centre of Tokyo, the JR Yamanote Line is very useful, as it runs through the main areas (Shinjuku, Shibuya, Tokyo Station, Ueno, Ikebukuro) is a circular fashion.
For many lines (including Keio) you can buy 11 tickets for the price of 10 (this is called "kaisuuken"), from (certain) ticket machines. Otherwise there are few discounts, and there are no discounts if you travel from A to B with one company and then B to C with a second company - you just have to buy two separate tickets.
There is a ticket giving unlimited travel on (ordinary) JR trains within the Yamanote circle line, valid for one day, which costs about 750 yen. This kind of ticket is called a "free kippu". You can buy this ticket from certain JR ticket machines at Shinjuku or at other JR stations. There are similar tickets for subway lines, but bear in mind that there are two subway companies and their tickets are not interchangeable.
In general, local trains in Japan are very punctual, clean, safe, and reasonably fast (and in Tokyo, much faster than the alternatives). Smoking is not allowed on trains, nor on stations except at certain designated areas. If you have heavy luggage, it is advisable to avoid the rush hour, especially as some smaller stations lack escalators. Luggage trolleys are not available, but almost every station has small storage lockers ("coin lockers").
If you need to enter the station platforms (e.g. to meet someone) then it is necessary to buy a platform ticket ("nyuujoken"). These cost 130 yen at JR stations in the Tokyo area and are available from some machines or from the ticket office.