Maybe I can help?
Well, there are a few I could recommend right off the bat. They arent too expensive, so you wouldn't be breaking the bank.
Semi-auto.
First (my first rifle), the Marlin 795. 125 with $25 mail in rebate. There arent many things you can do to this gun, but there are a few things available. You can get a dragunov stock, bullpup stock, etc off the top of my head. If you really dont care about customization too much, you cant go wrong. This has to be one of the best budget guns ever manufactured. We wont be able to get more than 10 rounds for these... :'(
Second, the Mossberg 702 Plinkster. Also a pretty cheap option. I really don't know much about it, but it seems to be a good rifle. Check it out. You're gonna be in the mid 100's I believe. 10 rd mag.
Third. Ruger 10/22. This isn't the most accurate gun out of the box, but it will do fantastic for most people. You can customize it to no end, there is just so many stocks, aftermarket sights, triggers and all that good stuff. The only problem (for us both) is that we live in states with high regulation. No mags bigger than 10 for us, but I beleive you can get good pre ban mags. No idea where. They are available though, this is an old gun. At my local shop, a basic model seems to go for about 200-230 bucks. Kind of expensive for what you want.
Bolt-action
Mossberg 802 Plinkster. This is the bolt-action version of the 702 plinkster. I've honestly heard nothing bad about this rifle. Seems like a good option. Again, not too knowledgable on the Mossbergs, but I'm sure this won't be more than about 175-200...
Marlin XT-22. Probably one of the best budget .22's on the market (for what it is). It seems to be extremely reliable when it comes to accuracy. For more information, check out "nutnfancy"s XT-22 video on youtube. It's worth the watch. I'll be picking one up myself soon. Looks fantastic and performs as well as it looks. My local shop has it for $225. The only thing I thought twice about was a stiff bolt. I worked it a bit in the shop, but it seems like it will loosen up quickly. A few hundred strokes before even shooting should do the job. Not that bad of a task. Looks like a good deal for a great modern looking rifle. I believe it's a 5 round mag? I think the 795 mag fits it, but in all honesty, I'm not sure. 7 rd mag.
Savage Mark II f (f being the basic Mark II model). MSRP is $214. The new models have the accutrigger, which seems promising, but to a veteran shooter, it wont make a difference. You can find a non-accu-trigger model for about $130-$150. I would recommend the accutrigger model, since you're relatively new to firearms. The Savage Mark II is a veteran in the rimfire world. It seems to be a good option and I don't think it will do you wrong. Was my original choice for a bolt until I found out about the XT-22. I beleive it's a 5 rd mag.
Verdict?
You want something relatively inexpensive. I would recommend the 795 simply because it's cheap and right now, I just dont feel like I want too much customiztion. When I do, a 10/22 will be purchased. But to start off, I would recommend the cheapest thing you can get. Now here, cheap doesn't mean crappy. The 795 is a great rifle, not only for the "price", but all around a solid gun. You can also see Nutnfancy's video about it on youtube. He hit the nail on the head with that review. As for ammo, my 795 like Federal Bulk (Value pack version) the one you can see at walmart (in a redish box). BY THE WAY, the 795 can't shoot hyper velocity, I forget what it is, but it wears out... I believe it is a plastic recoil buffer (I think it's called) is to blame. As for a bolt, I plan on getting the XT-22, but if I didn't, I'd go with savage. You said you didn't want to spend $400, but if you do spend that much, you could end up with 2 of these rifles that I mentioned and also a few thousand rounds of ammo depending on how expensive the guns you buy are.
I just hope I pointed you in the general direction of something you may want to buy. Anyway, good luck. You can message me if you need to ask anything. Again, good luck. And most importantly, have fun!