I have something close to this model and I wouldn't be afraid to tackle something like what you're doing with it.
https://www.amazon.com/dp/B0049ZFUK2/ref=cm_sw_r_cp_apa_fab_.L1AFbHXYRMKT
The plunge attachment will be helpful, particularly for getting the cut started, but I mostly use mine for flattening rough cut slabs.
this is slightly above budget but I think you'd find the plunge base incredibly useful.
This is the one I bought. Super happy with it. I mounted it upside down on a table and turned it into a table router. works great.
I bought this one from Dewalt some time ago. I've yet to find a use for it but it too has the option for plunge/fixed-based adjustment options. Costs a little less...
I may just bite the bullet and buy this: http://www.amazon.com/dp/B0049ZFUK2/ref=wl_it_dp_o_pC_nS_ttl?_encoding=UTF8&colid=2G5YCHRW0LJRJ&coliid=I2VMZMVGZIV7LS
Any thoughts on the compact routers? It's obviously not for heavy use just occasional projects.
Here is the plunge router and jig I use. The jig changed a little but when I bought it it didn't come with the screws to fit that Dewalt but I go them at bLowes. Just take the jig with you to make sure the screw heads sit flush when you put them in. It is easy to use though and makes friggin great circles.
You could stick with the DeWalt line and pick up a trim router: https://smile.amazon.com/DEWALT-DWP611PK-Torque-Variable-Compact/dp/B0049ZFUK2/
This is a great kit that includes the plunge base.
If it is his first router, I would recommend getting the DeWalt 611PK kit. It is a compact router, but it is pretty powerful for its size. I bought the Bosch recommended in another comment as my first router, and looking back, I wish I had purchased the DeWalt first. I find myself using the smaller router far more than the bigger one. Plus, the entire kit is around $160, so you have plenty of money to get a MCLS bit set and still be way under budget.
Just my 2 cents.
DEWALT DWP611PK 1.25 HP Max Torque Variable Speed Compact Router Combo Kit with LED's
Ultimately if you get serious into woodworking you are going to end up with several routers, a 3+ HP router table, at ~3 HP handheld, a 1.25 HP handheld, and a trim router. It all comes down to what you are doing with them. The router table is your work horse that you use for 90% of tasks, except for the few operations you can't do. The big routers are unwieldy to handle so you really have to make sure they are well supported and stable or then can get away from you. For larger bits you need the 1/2" shank and the extra power they provide but for smaller work you give up control and it's a nuance to use them. For those sorts of projects a smaller 1.25 HP compact router is the better option. You can't use the larger bits up much easier/safer the handle and more control. They use a 1/4" shank. For really fine work that you want to do free hand, say routing out waste for inlay, the trim router is what you want.
You want definitely want a plunge router. That allows you to start/stop in the middle of a piece by plunging down into it. I don't know that I've ever bothered to switch a router to fixed base. Variable speed control is also a must, you will get a much better cut if you have the correct speed for the bit/material you are working with. As far as as shank size you generally want 1/2" as it's safer, so if there is an option between 1/4" and 1/2" for that bit you'd usually go for the the 1/2". A lot of the smaller bits are 1/4" only, larger 1/2". You can use a 1/4" shank bit in a 1/2" router but not the other way around. So keep what bits you need in mind when you pick a router. Another important aspect is the router bits you use. Cheap bits are not worth it. A lot of people will say buy a cheap set then upgrade the ones you use regularly. I did that, not a great idea. The quality of the cut is poor and they don't feel safe. Do you want a cheap made in China bit spinning at a high RPM in a tool you are holding? No thanks! I get 1000% better results from good bits, smooth cuts and no issues.
For the sound of what you want to do I'd suggest a smaller compact router to start (then a router table /w dedicated 3+hp router motor). I like this one:
As far a bits go my top choice is Whiteside. The Freud quadra cuts (larger 4 winged bits) are super nice as well. There are several other good brands out there too though. I'd buy bits on an as needed basis as they are expensive for good ones.
DeWalt, Bosch, and PorterCable are popular router brands.
I picked up this Milwaukee BodyGrip for $90.00 at a pawn shop. Barely used, still had the case and accessories.
I bought it for a router table application and the occasional hand work, so the plunge wasn't a necessary feature.
If you're not using it in a table, the DEWALT DWP611PK 1.25 HP Var. Speed Combo Plunge kit is a very versatile option, with plenty of great features and 5-star reviews.
What you don't want to skimp on is the router bits. A good bit will last with intermittent use; and more importantly; won't shatter on you.
Freud, Rockwell, Yonico, MLCS, Whiteside...all good brands. The Diablo line at Home Depot will do in a pinch, I suppose. You can save a few bucks by getting a basic set of commonly used bits.
I wanted to buy a hand tool, but I couldn't see the logic in doing so when I could get a powered router for the same price.
https://www.amazon.com/gp/product/B0049ZFUK2/ref=oh_aui_detailpage_o06_s01?ie=UTF8&psc=1
I was debating on this one and the makita that i saw was recommended. But this one ended up being cheaper on prime day.
Edit: I have never used one of these things. What kind of bits are ideal for woodworking?