We are newly married couple with no kids.I hope with 3000(after tax) we can have a decent lifestyle.

Assuming 2 adult + 2 kids... 

$900 Rent
$200 Utilities (light/water/trash/internet/cable) 
$200 Car Gas+Insurance
$800 Grocery
$400 other stuff
--------
$2500 Total monthly expense

With $36K per year household income for married couples + 2 kids, NO income tax. In fact you may qualify for some money from IRS.

But "Health Insurance" could be huge additional expense if not provided by your employer.

if $3000 per month is after tax and health insurance, puts you in lower middle class.

if $3000 per month before tax and health insurance than puts you in poor category.

Now if you are single, that change all these numbers. Your health insurance will be way cheaper but your income tax obligation will a bit higher, you may share apartment with other singles and a little less other expenses.

depends if you are moving with/without family/kids. if this is your first job, age, lifestyle, etc. in a nutshell, 3000 should be doable if you want to enjoy a reasonable lifestyle for a family of two-three. Car is not must but close to must because it makes your life a whole lot easy and efficient. gas is not that expensive, grocery should be around 300. Utilities are under 100, If you keep your rent under 1000, i think you should be fine for the most part. feel free to ask questions.

Cost of living is good in Nashville compare to other Metro cities. To give some idea rent for 1 BHK - around 800. 2 BHK - 1200. The saving  depends upon the lifestyle but my personal opinion is with 3000 there wont be much savings. Hope this helps.