Can I replace my bath fan with one of similar size?
If you're looking to replace your fan here's what you need to consider


Sound - Measured in sones, a lot of older square fans are fairly loud. Those are 4-5 sones. A lot of newer fans are a lot quieter. I was able to install a 1 sone and it's so quiet you can't hear it outside the bathroom. The boxes typically display this prominently
CFMs - Cubic Feet per Minute. This measures how much air the fan can move. The small, noisy fan probably moved 50 CFMs, if that. My new fan is larger, but moves 90 CFM. With the new fan, I can take a full-on hot shower and have the small bathroom (maybe 50 square feet) de-humidified in about 5 minutes.
Ducting - If you can go into your attic, you should. Try to figure out where fan sits. Sadly, it's very possible that the fan does NOT duct outside. This was what I found in my bathroom (the insulation was filthy from years of dust dumping). This is a problem if it doesn't (can cause mold to grow), and there's a good chance that, if they did duct it, it's not insulated. Especially if you live further north, you want this line insulated (warm, moist air into a cold zone will cause condensation). Most ducts and vents are 4", but if you go above 100 CFM, the duct size tends to go up to 6" line. You can buy insulated flex ducting (check the ducting area of your hardware store, as fans are sometimes sold near light fixtures).
Venting - Sadly, the only roof vent I could find was in a kit that came with uninsulated flex line. It's a compact 4" duct, and can be cut with either a large hole saw, reciprocating saw, or jigsaw. Be sure to buy some appropriate roofing or outdoor caulk. DO NOT USE BASIC SILICONE ON YOUR ROOF. If you're scared of cutting a roof hole, you can also buy a soffit vent, but be aware that the warm, moist air will go out the vent and some might come back up through the vents of the soffit back into your attic, where it can cause mold. If possible, vent out of the roof.
Wiring - Honestly, if you've already got a fan wired in, you can keep the same wires. Just be aware that it's important to ground these fans. Your previous fan may not have one (or they may not have attached it). Don't skip that step. Make sure it's grounded. Also, consider putting in a timer switch. That way your family and guests can set the fan to run after they're done with the bathroom


Nutone actually makes some fans that are designed for easier DIY replacement. Some of their models can be replaced entirely from the bottom, instead of trying to get to it from the attic. This may involve some drywall cutting to accommodate a larger form factor, but they keep the height of the smaller, traditional 6" x 6" (typically 3"). You keep the same wiring and venting.