Hi Mike, welcome to the fun. $125 isn't much of a budget for a router. If you can, I suggest you check into a Bosch 1617 EVSPK kit, refurbished, from CPO. They've been in the 160 range and have a good reputation here. The kit includes both the fixed and plunge bases. The fixed base can be placed under a table (shop made is fine), which is pretty much the safest way to do many tasks. My first router was pretty cheap and didn't have a plunge base. I was happy when I got rid of it for about $60 including a small table. I was OK, but not much power.
You can get a new Bosch 1617 with the fixed base only, but before long, you'll want to have the plunge base for cutting stopped dados (a groove that begins and ends within the edges of the material, and that will require a plunge base. If you buy the plunge base later, you'll pay a lot more than getting the kit in the first place.
You are within the price range of a trim router, but they only take the 1/4 inch shank bits, and with only about 1hp, they are limited in the projects you can do with them. So whatever you buy, you want to get both 1/4 AND 1/2 inch collets, which is the nut you tighten to hold the bit in place.
For general bits, I'm happy with Freud bits, which you can get at Home Depot. You'll probably start out with what are called roundover bits. Most of us have the 1/4, 1/2 and 3/4 inch roundover bits. They're the ones you'll be using all the time. Since they're small and light weight, you can probably do fine with the 1/4 inch shank models. You'll be making several passes with these, not hogging out the whole roundover in one pass. I pretty much limit my cuts to about 1/8th inch cut per pass.
If you decide to get the Bosch machine, you can use the fixed base for table use, but you'll want to order a key (see pix) so you can adjust the bit height from above the table. It's cheap.
Making your own table means taking a piece of flat ply, cutting a hole in it and securing the router base to the bottom. Many people here have been using this simple table suspended over a couple of sawhorses for years. If you want a simple, you find a really flat 2x4 and clamp it to the table. hint: cut out a spot in the middle for the router bit to fit in.
Routers produce a lot of sawdust, and if you can, I suggest you use it outside until you have some dust collection in place. Sawdust is very nasty stuff that can ruin your breathing and health. Get yourself a dust mask (3M makes an inexpensive model with a one way valve. Wear it at all times, particularly if you are working in an enclosed area. Sawdust hangs in the air for a LONG time, and sweeping it up stirs up the finest particles, which are the most dangerous. Working outdoors will help, but do get a mask. Some of us use a 20 inch box fan with a 20x20 filter taped to the front. Let it run for a couple of hours at a time to reduce the suspended air. Hang it from the ceiling so it doesn't blow on the shop floor and raise more sawdust than it catches.
I also suggest you watch videos of routers in use. There are hundreds of them on YouTube. Among the best are by Marc Sommerfeld, who sells router stuff, but his videos teach simple and effective technique (he was a cabinet maker before starting his company).
I'm sure Stick will post his pdf collection and safety information shortly. Well worth reading. I also have a long post on the 17 plus things that helped me get going in woodworking, it may help you avoid buying stuff you really don't need. Here's the pdf.
Woodworking is a great hobby, glad you've found us, there are hundreds of years of woodworking experience around here, and we all love answering questions. We also have a little fun now and then.
