bell_d
Hello and Welcome to the Forum!
What a nice locomotive to start with! I agree with your Mothers' hint about seeing it run again.



Click these link's for info about your locomotive.

http://www.postwarlionel.com/cgi-bin/postwar?ITEM=1862

The 1862 locomotive was issued with O27 set's,  first released in the #1800 the General Frontier Pack in 1959.
It will run on O track as well as O27.  Originally it would have came with tubular track. It would be your choice to go with the original track or  modern track.   A used transformer should serve your purpose,  I'm not sure where you're located or if you have a local hobby shop available.  If you do have a LHS check with them for used track and transformers, also check Craigslist, local papers, etc.  I don't recommend ebay or other online auction sites, the shipping/handling may make the used items cost as much or more than new.
Many people buy starter sets, then replace/upgrade the track/transformers that came with the sets, so keep your eyes open for them.
IF you have difficulty locating used track or transformer, contact me at the email address listed in my profile.

Happy Rails!
Dave