Yeah, you don't need Legal Zoom to file a DBA (sole proprietorship). Fill out the form and file it yourself at with the County clerk. Do a Google search for "Brown's Lawn Service" to see if there is anybody near you using that name. Get a few certified copies as you will need to give one to the bank in order to open an account- "Brown's Lawn Service".
Now, you could conceivably not even do any of that if you intend to use your real name and have customers make checks out to "Bob Brown." But it is best to file a DBA (doing business under an assumed name) even if it is yours and open a separate business checking account. You will want to then request a TIN (taxpayer identification number) from the IRS and, most importantly, a sales tax number (often it's the TIN the feds gave you) from NJ department of taxation because you will have to collect and remit sales tax. And believe me, you don't want to get on the wrong side of them so get that right.
As for your tax question, if you are self employed in business you file a schedule C along with your 1040 on April 15 to report a profit or loss from a business or profession. The IRS doesn't really care if you are a DBA or use your own name as long as you report your income. If you make enough you will have to make quarterly "estimated tax" payments. Deductions for and depreciation of assets depends on what they are, new or used and other factors. That's why you should have an accountant to prepare your taxes at the end of the year. For the few hundred bucks it's worth every penny to stay out of trouble and do things right. You will also want to keep accurate business records, not just throw receipts into a box and give that and your checkbook to your accountant like some idiots do. I recommend an accounting program like Quickbooks. For a LCO the setup is simple, all accountants know it and you can just give him a copy on a thumb drive at tax time.