DSLR camera speedlites work in camera's auto mode?

I'm looking for a cheaper priced speedlite that will fully work on my nikon d3000. I notice that NEEWER TT560 looks good, but that it will only work in my camera's MANUAL mode… Are there any affordable speedlites that will work in the camera AUTO mode or am I doomed to work in M mode if I get a speedlite of any kind… Unless I spared an arm and a leg to get a more expensive kind… Thanks in advance.

You don't save money by buying something you need to replace with what you should have bought to begin with.

Firstly, you should think of learning to use both your flash AND your camera in manual modes as 'dooming' you to anything. It's what you should be learning to get the best of out of your gear.

You can buy an all singing all dancing Nikon TTL speedlight, but you are limited to what you can do with it off camera (which is the best place to have a flash) because your Nikon model does not have the Nikon AWL/CLS commander function.

Given that that's the case, you might as well go manual.

I use a Yongnuo YN560 (about £50) plus RF602 radio triggers. Despite having Nikon speedlights and a camera that can use CLS I much prefer to go fully manual & have full control over my exposures.

People get the wrong idea about manual flash - it's not hard to learn. Have a look at the Strobist site for more info on off camera flash - Lighting 101 & 102 archives are the place to start;
http://strobist.blogspot.co.uk/2006/02/welcome-to-strobist.html

This is a shoot I did using a couple of YN560's and RF602's;