PNG is a 100% replacement for GIF files and is supported by all web browsers you are likely to encounter.
There are very, very few situations where GIF would be preferable. The most important one is animation--the GIF89a standard supports animation, and virtually every browser supports it, but the plain old PNG format does not--you would need to use MNG for that, which has limited browser support.
Virtually all browsers support single-bit transparency in PNG files (the type of transparency offered by the GIF format). There is a lack of support in IE6 for PNG's full 8-bit transparency, but that can be rectified for most situations by a little CSS magic.
If your PNG files are coming out larger than equivalent GIF files, it is almost certainly because your source image has more than 256 colors. GIF files are indexed to a maximum palette of 256 colors, while PNG files in most graphics programs are saved by default in a 24-bit lossless format. If file size is more important than accurate colors, save the file as an 8-bit indexed PNG and it should be equivalent to GIF or better.
It is possible to "hack" a GIF file to have more than 256 colors using a combination of animation frames with do-not-replace flags and multiple palettes, but this approach has been virtually forgotten about since the advent of PNG.