A good bolt action will get you 400+ yards easily.
Given your budget I would say the following rifles are very good. Keep in mind I am going to break the rifles down into two catagories. 1) will be out of the box with no mods 2)out of the box and able to modify.
If you aren't looking to upgrade the rifle down the line...the following rifles are very good.
Howa rifles are great shooting rifles...only "issue" is that they use metric threads, so many gunsmiths won't work on rifles that are threaded with metric threads.
Steyr rifles are very solid as well. CDNN had them for $600 price point, but they sold out of the Steyrs.
Tikkas are nice rifles as well...they will have the smoothest action from any factory rifle. Tikka is starting to gain more aftermarket support, so you can make some changes down the road. I heard a rumor that Beretta (owner of Tikka/Sako) were going to discontinue importing Tikkas...probably because the Tikka line would be competing against the Sako A7 line.
Rifles that are inexpensive, but are capable of being worked on by yourself.
Savages rule for this...avoid the Edge/Axis line because they are throw away rifles and not able to be modified. I am referring to Savage/Stevens 200 rifles. Savage rifles will shoot out of the box, but if you want a good custom barrel, you can get one for around $300. You can buy the needed tools for around $100 and that will allow you to swap out your barrels. What that means to you is say you get a 308 in a SA(short action) you can add different calibers to your action. You can go up to a 7WSM using the short action, or you can go down to a smaller caliber if you want to as well.
Marlin XS7/XL7 are clones of the Savage since they use the same nut system as the Savage. The Marlins are suppose to be good shooters out of the box as well. The big disadvantage of getting a Marlin over the Savage is that Marlin is owned by Cerberus and they are closing down the Marlin plant. Looks like all warranty issues will go to Remington in NY. I personally do not like how Cerberus is closing down a bunch of plants from companies they purchased the last few years. I'd be concerned that one repair department at Remington will be able to handle so many different firearms at one location.
If you go with the Savage...do yourself a favor and replace the cheapie factory stock. Now some Savages come with a McMillan, but most come with a horrible stock. If you want a budget stock...take a look at the Bell & Carlson Medalist stock...decent stock for around $225.
Take a look at EGW 20MOA base for your scope base. They are a local company to me in PA and have a good reputation for producing quality parts. Get the 20MOA base since it will help your optic reach out further (if you get an optic that doesn't have a lot of elevation travel). The Burris Xtreme rings aren't bad for the money...they are aluminum rings, but work well.
That leaves you with about $700 on optics. I personally have been going with nothing but Vortex optics lately. Their glass is good and their customer service is amazing. I bought a new Vortex Viper and noticed that it had plastic inside of the glass sticking out on the corner of the scope. I called CS and they sent me my replacement scope and a return label to return my defective optic. This type of service is unheard of these days. I can tell you that Streamlight (former employer) would never send out the replacement prior to getting the defective unit first. Vortex was very trusting considering that they sent me my replacement before I sent in the defective unit. This is the reason why Vortex is getting such a loyal following in the shooting community. I like my Viper a lot and I bought it a few months ago on clearance since it was a discontinued model (the reticle wasn't selling that well) for only $250 shipped. I wished I would have bought more of those scopes at that price. I would highly recommend whatever scope you choose that you at least get an optic with a 30mm tube. It will give you better light transmission. I'd say get a 34/35mm optic if you can afford it.
I have owned expensive rifles and I can say that my Savage that I rebarrelled myself has outshot my Remington 5R. I like being able to swap calibers myself and not have to pay someone to do the work. Take a look at Savage/Stevens 200 and the Marlins.
Tools you will need.
Nut wrench
http://www.midwayusa.com/viewproduct/default.aspx?productnumber=288765
Action wrench
http://www.midwayusa.com/viewproduct/?productnumber=710783
Headspace gauge
http://www.midwayusa.com/viewProduct/default.aspx?productNumber=491140
Ok, so it's more than $100 in tools, but great if you want to swap barrels/calibers and save money by DIY.