With regards to the "lifting" process, every city is different.
If you want to do it without an attorney, usually you must post a bond in an amount equal to the fine. For instance, if the current fine on your speeding ticket is $175, and the failure to appear fine is $200, you must put $375 on deposit with the court for them to lift the ticket. You can then set a court date, at which point you may be able to negotiate a lower fine. Alternatively, you can just plead guilty and pay the $375, at which point the warrant will be gone and the tickets will be disposed of. A major caveat of this is that a speeding ticket can affect your drivers license and your insurance rates.
If you hire an attorney, the city may lift the warrants upon the attorney's sending notice to the court that he is representing you. Not all courts do this (actually, most don't), and you should check with the court to determine its policy.
More likely what your attorney will do is post an "appearance bond," which states that the attorney guarantees your appearance, or else the attorney will owe the full fine to the court. Practically this means that the attorney is promising to show up for your court date, because in municipal courts the attorney may appear on your behalf. An appearance bond usually does not require any money to be put on deposit with the court, but an attorney usually will charge for this service.
The type of attorney you need in this situation is one who dedicates some portion of his practice to traffic tickets. To find an attorney, start by asking your friends and family if they have worked with anyone on a similar issue, or if they can suggest someone to you. There's always the phone book, though that can be overwhelming and it is hard to determine whether someone's nifty Yellow Pages ad means the person is actually a good attorney. You can also use the Google, or search through Avvo.
Know going into this that the attorney will charge for his services, and that if any fines are assessed, you will be responsible for them in addition to the attorney's fees.
This answer is based upon the limited information provided in your question, and is not intended to be legal advice nor to create an attorney-client relationship. For more information on your particular situation, please contact a local attorney.
Easiest thing to do is call the court, tell them you want to take care of it and see what they offer. Some will do it by credit card, mail, even getting charges dropped and getting a deferred. It can't hurt to try the simple way before having to start hiring attorneys in this case, but do it soon, least you get pulled over and arrested.