Should I take the first offer my insurance gives me for my car ?

You probably won't get another offer, just so you know.  You can also ignore all the get-a-lawyer answers, because they're dead wrong.  Unless you were injured in the crash, no lawyer wants to take your case.  No money in it for them unless you're hurt.

I've handled over a thousand accident settlements myself, and I've yet to see a single person who thought they were paid what (in their opinion) their vehicle was actually worth.  I've seen people curse and swear, provide pages of for-sale listings for comparable vehicles, threaten court action, and write letters to everyone including (possibly) JollyOldSanta.com to register complaints.  And only one of them got any success in their pursuits.  Here's why...

When insurance companies offer a settlement, they've already done the necessary homework.  They know that any claim could be audited by whoever regulates insurance companies where you live (there's one in every jurisdiction), and they know that any claim could end up in front of a judge at any time.  They also know that if they don't have all their documentation in perfect order every single time, the regulatory authority and / or judge will absolutely crucify them.  Whether the claim is for 20 bucks or 20 million, they're expected and required to follow due diligence without fail.  That's what they do for a living every day, and that's why they don't get scared when customers threaten to get a lawyer. 

But give it a try anyway, because there's no reason you can't ask.  They might kick it up by a hundred or two, because your vehicle had (just for example) an upgraded stereo or those cheesy pimp-lights or whatever.  If you ask nicely and provide them with some sort of coherent reason, the worst thing they can do is say no.  They won't blame you for asking.

How did that one guy do it, you ask?  He hired his own independent appraiser to do a complete without-prejudice (that's a legal term) assessment of the vehicle's value, and he presented it to the insurance company.  The insurance company either accepted the second independent appraisal, or they accepted that it wasn't worth spending $25,000 to prepare for a trial over a few thousand bucks in settlement.  And that's just the cost of preparing for a trial, things get expensive after that.

Call a attorney never take money from an insurance company without having a lawyer 

Insurance adjusters only exist to cheat you out of money.

Merely inform the adjuster your neck really hasn't felt good ever since the crash.

You would be amazed how much they will offer to settle out of court

yes , thats all you will get , cost more to repair than payout .. They give you market value , not replacement value . 

You won't know what is "fair" until you do some searching to see what like cars in like condition are selling for in your area.

I just had a car totaled and after doing some comparison searching I found the first offer was more than fair, so I grabbed it.

The first is to avoid taking the first offer made. According to Nolo, Sutliff & Stout, and Findlaw.com, an insurance adjuster will often make an extremely low first offer to determine whether you know how to negotiate or understand the value of your car.

If the settlement is fair, take it.  Otherwise you would have to show that the offer is not adequate.  You can go to Kelley Blue Book, www.kbb.com, and get your own estimate of the value of your car.  Licensed auto appraisers in my state, PA, are bound by state law to make legitimate offers.  (retired claims adjuster)

You're supposed to get two body shop estimates on your own! You don't just let the other drivers insurance company make an offer. You shouldn't even be talking with them with out an attorney.

You will always get more if you hire a lawyer. But only if you were injured.

Assuming your Dad is an attorney and has PROOF that the "price" is reasonable (a very odd use of words), take his advice.  If he's not, evaluate your damages and the "price."