Clone your DNA via bacteria or PCR using ddNTPs.
When DNA is cloned, it essentially replicates using deoxynucleotide phosphates (dNTP) added onto a primer. Dideoxynucleotide phosphates (ddNTP) have hydroxyl groups removed at two places: the 2' and 3' carbon. The first hydroxyl group removed at the 2' carbon makes it a deoxynucleotide (which makes DNA different from RNA). The second hydroxyl group removed at the 3' carbon makes it so that another dNTP cannot be added on. Recall that NTPs add on at the previous NTP's 3' hydroxyl group.
If you add a small amount of ddATP and clone the DNA template several times, the DNA will stop at several places where dATP is supposed to be added on. You can find out where the ddATP stops by running the DNA through gel electrophoresis. If you do this for each type of dNTP (A, C, G, T) you can find out which dNTP is next in sequence based on where it runs on the gel.
For instance, you run 4 gels, one for each type of dNTP. You will get something that looks like this:
The gels run from 5' at bottom to 3' at top, so going by the gel above you can see that your sequence is:
Clone your DNA using ddNTPs that have fluorescent tags, then analyze on electropherogram.
The new method of sequencing DNA uses the same principles as the old one, but with new technology. Essentially, you still use ddNTP to prematurely stop DNA replication. However, now that you have fluorescent tags and an electropherogram, you can dump all the four different ddNTPs in one batch and run it all at once. In the old method, you had to run each ddNTP separately and run each on a gel. Now, you can dump all four in one cloning reaction and run it through the electropherogram. The electropherogram is essentially a gel that can differentiate among the different ddNTPs using their different fluorescent tags. This is faster and more accurate than the old method.