I don't know a ton about lever guns so I'm going to restrict my comments to bolt action rifles; other than to say this, you can get some pretty sweet deals on lever action rifles in 30-30 or even 32 special if you are willing to go used. My brother got a Winchester 94 from 1967 in 32 special in unfired
condition last year for somewhere around $500.
If you choose to go with a bolt action and the budget is under $1000 it's very hard to beat the Tikka T3 line of rifles. The T3 Varmint is available in 204Ruger which will take care of any varmint you have in mind at the range you want and it is, of course, available in 223R. You can see it here Tikka T3 Varmint
and there are several other variations of the T3 if you want a wood stock or a thin barrel. The Tikka will shoot extremely well and the action is as smooth as silk soaked in hot butter.
Another excellent rifle right out of the box is a Howa and they have several varmint models and hunting models available for the $500-$600 range. If you want to modify the rifle there is not limit (almost) to what you can get for a Remington 700 and Savage makes a nice rifle for reasonable money as well.
For plinking or shooting short range there isn't much as fun as a good ole Ruger 10/22 and if I'm not mistaken you can get a 17HMR barrel for it for reasonable money.
Sorry for being long winded and I hope this helps.