Attorney Butterman has provided you with excellent information. I will only add to his answer that there is some very specific language that needs to be included in a Trademark Assignment document (the contract that sets forth the terms of the sale of the trademark) in order for the assignment to be a valid sale. The document should also include language to ensure that you fully step into the shoes of the seller, including the right to bring a trademark infringement lawsuit against anyone who is infringing the mark at the time of the sale.
Therefore it is extremely important that you engage an experienced trademark attorney to either prepare the Trademark Assignment document for you (or if the seller has prepared it, to review the document to ensure that it is valid and contains all the necessary and helpful provisions you would want). This is not something to try to figure out on your own or ask an attorney who does not specialize in this area to figure out.
Bottom line - if you are going to spend any money to acquire the mark, you'd better make sure in advance that you are truly getting your money's worth and that everything is done properly.
This answer is for general information purposes only. This communication does not constitute legal advice, nor does it form an attorney-client relationship.
The short and dirty answer to your question is YES - you have the same rights as the trademark seller and can sue infringers - but with one HUGE caveat:
The seller of the trademark has to have been continuously "using" the mark in Commerce when it sold the mark to the buyer and the buyer has to continue "using" that mark in the same manner - applying it to the same or similar goods/services and continuing to use it in Commerce. If the mark seller had stopped using the mark, it may have been deemed to have abandoned the mark and the buyer merely bought an abandoned mark. The assignee (buyer) of a trademark steps into the shoes of the assignor (seller) of the mark.
When you refer to trademark "rights," you refer to the ongoing use of the mark in Commerce which means the continued application of the mark to the goods or insertion of the mark in advertisements for the services in what is considered ordinary commerce that Congress can regulate. Trademark rights do NOT refer to a registration or registration certificate which may be valid in the records of the US Patent and Trademark Office because a regsitration for a mark in Commerce that is not used in Commerce is effectively a legally abandoned mark, although a court or the Trademark Trial and Appeal Board would have to determine that this registered mark was legally abandoned. And there are a number of different rules about how long the mark needs to have been out of use to be abandoned, what is use, what is non-use, etc.
You really need to run your specific facts by a privatey retained attorney attorney to receive proper, accurate advice on this issue. In this public forum, you do not want to reveal any specific facts and we cannot properly advise you without the specific facts and because we cannot form a proper attorney-client relationship in this forum which would retain your attorney-client privilege.
I hope that helps. Good luck.
Alex Butterman is a trademark attorney with Staas& Halsey LLP (http://www.staasandhalsey.com), a Washington, D.C. IP boutique law firm. Alex is admitted to the bars of Washington, D.C., New York and New Jersey but, unless otherwise specified, the answer is intended to be general enough to apply to any U.S. state and based primarily upon his knowledge and experience with applicable federal laws. The opinions expressed are those of the author and do not necessarily reflect the views of his firm, Avvo or other attorneys. This answer is for general information purposes only and is not intended to be and should not be taken as legal advice. No attoreny-client relationship or obligations are established herein, although consulting an attorney to discuss your specific situation is strongly recommended. This is especially true of trademark law and probably any area of intellectual property law because TM law is so fact-specific and full of esoteric nuances and exceptions, that more often than not there does seem to be harm in handling a trademark matter without consulting an experienced trademark attorney.
In addition to the other excellent advice in the previous answer, your rights would be set out in the agreement of sale for the mark. You should get the advice of an intellectual property attorney to ensure that you are purchasing everything that you require to have exclusive rights to the use of the mark. Make sure you are actually purchasing the mark itself, not merely a license to use the mark.
The TM office site will not update itself to indicate that a trademark has gone abandoned. There is no way the TM office could police every single TM that gets registered. So the only thing that you can tell from the TM site is that the last statement of use was filed. The mark could still have gone abandoned as it can occur in as little as three years of non-use, if the original business does not have a pretty good reason for having not used it.
So again, to make sure you don't buy a pig-in-a-poke, you should consult with a real IP attorney to evaluate your particular circumstances.