If it's your trademark you can use it as you see fit, provided you don't infringe on the rights of others.
If your enterprises are separate entities -- and they probably should be -- consider the wisdom of licensing the trademark by the entity owning the trademark.
By definition a trademark indicates the source, origin, affiliation, connection, association, sponsorship, or approval of goods and services. So, in your example, using the same "Lightning" trademark for the sporting goods store and the skating rink simply tells consumers that the source is the same for both services (or there is some relationship between the sources of the two services.)
No, you could not infringe your own trademark unless you set up separate companies. Even then, your attorney (and you need to hire one to review the specifics) could prepare suitable license agreements to legitimize the usage and protect the mark. Get an attorney.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
This question brings up alot of questions concerning trademarks and their registration. At their most basic, a trademark is merely a symbol used to identify specific goods or services and to differentiate them from a competitors. This symbol could be a word, a design, a color, a shape or anything else. What it isn't .. is the name of the company, or even the company's logo. While those things can, in some circumstances be registered as a trademark, a mark that is merely descriptive or identifying of the company is often decline registration. Assuming your symbol is registerable as a trade or service mark, it is not registered according to your business, though your business might listed as the owner. Marks are registered according to the product or service they are used to identify according to a complex listing of classes maintained by the US Patent and Trademark Office. Each mark has to be registered to at least one class of products or services, and many are registered to multiple classes. Each class listing is another registration fee. The whole point of this complexity is to prevent confusion in the mind of the consumer as to the origin of a product or service. In you example, Lightening Sports, assuming it is registerable, would be registered as a mark denoting sports equipment. If you were to open up a skating rink called "Lightening Sports", obiviously there is no conflict. If the same company that owns that mark were to use Lightening Sports as denoting a certain sporting venue, ie,., a skating rink, then it would behoove you to register that mark in the class catergory that included sporting venues. If you didn't, then anyone may be free to open up a skating rink using the logo to denote a sporting venue as that is a completely different product class, and unless registered in that class, would remain unprotected. Assuming the symbol "Lightening Sports" is registered in all appropriate classes, then the owner of the mark would be free to license its use to any other company. That is what you would do if you wanted to open your skating rink using the same symbol. I'm trying to avoid using the word "logo" or "company name" because, as I said earlier, that is not necessarily a trade or service mark that is capable of being registered. Finally , there is the question of Federal versus state registration. Generally interstate commerce is required to permit registration with the U.S. Patent and Trademark Office. A Skating Rink located in a single state might be better off registering its mark with the State. In Georgia, state marks are registered with the Secretary of State and state law provides similar protection to mark owners in the state of Georgia on a intra-state basis. If the trade or service marks is an important matter with your business, then you really should meet with an attorney experienced in the registration of marks. It is a complex area of law. Good luck.
As my colleagues have stated, if both marks are owned by you, there is not an infringement. I would also agree that a separate entity should own the Lightening Rink and that a license agreement be provided from the original company to the new company allowing them to use and register the mark.
You will want to consult and utilize a contract attorney with experience in trademark licensing to assist with this.
Best of luck.
The information provided in this answer is for informational purposes only and should not be construed as legal or expert advice which can only be obtained from appropriate professionals. Additionally, this answer does not create an attorney-client relationship.