Denver International Airport: Rent a car at the airport; it is the genuinely practical option. Leadville is roughly 2.5–3.5 hours away in good conditions, generally via I-70 and US-24/CO-91. There is no dependable direct airport-to-Leadville public-transport transfer worth building a trip around; allow substantially more time if using a patchwork bus connection, and do not count on it for a late arrival or winter travel.
Eagle County Regional Airport: A rental car is again the sensible transfer. Allow roughly 2 hours, longer in snow, using mountain roads via the I-70/US-24 corridor. It can be a useful airport for a western approach, but it is not a plug-and-play shuttle destination for Leadville.
Lake County/Leadville Airport: KLXV is a general-aviation airport, not a commercial gateway: there is no scheduled airline service. It is exceptionally high—about 9,934 ft—and mountain flying preparation is essential. If you arrive in your own aircraft, town is a short drive; arrange a car in advance.