Train is the most reliable way of getting to Lipetsk. 2 overnight trains go from Moscow Paveletsky Station on daily basis, other train connections are with Saint Petersburg and Sochi. One way 2nd class ticket from Moscow to Lipetsk costs about 2000 RUR / 50 EUR.
Lipetsk has good bus connection with Moscow. Several private operators provide 7-10 daily trips from Paveletskaya metro station of Moscow to Lipetsk train station and back. Expect 8 hours to go, 600-700 RUR / 15-18 EUR to pay and loud video in the coach for all the way. Bus collisions are regularly reported on southern directions, so be attentive taking this risk.
Depending on traffic, the road to Lipetsk can take 5-8 hours from Moscow. Take M4 road to Rostov on Don and turn left to Lipetsk soon after Yelets. Alternatively, turn from M4 to M6 road after Kashira, then turn right to Lipetsk after some 200+ km drive. This second way is shorter in kilometres, but worse in road quality and traffic.