Sources for authentic recipes

If you're looking for books, you could try "Roman Cookery" by Mark Grant or "The Classical Cookbook" by Andrew Dalby and Sally Grainger for ancient Roman food, or "The Philosopher's Kitchen" by Francine Segan, which combines ancient Greek and Roman cuisine. Many of these recipes are derived from the works of Apicius, but are not solely based on his writings. "Roman Cookery" has been praised for getting beyond just Apicius (the best known resource on Ancient Roman cuisine), but also may be harder to find than the other two.

For Medieval cooking you might try "The Medieval Cookbook" by Maggie Black, which focuses on English and French recipes, or "Pleyn Delit" by Butler, Heiatt, and Hosington. Black's book was inspired by "Pleyn Delit" and the two books are somewhat similar, so you might want to only choose one, depending on your interest. Another good choice might be "The Medieval Kitchen" by Redon, Sabban, and Serventi (translated to English by Edward Schneider), which focuses on French and Italian cuisine in the era before New World ingredients were introduced, which sounds like what you're looking for.

All of these books include a lot of history and discussion of ingredients and cooking techniques of these eras, and often include original texts so you can judge for yourself how close the interpretations are to the original recipes.

On the web there are a number of sites devoted to Apicius and ancient ingredients and recipes.
For Roman you might browse this page:

http://www.cs.cmu.edu/~mjw/recipes/ethnic/historical/ant-rom-coll.html

or for Medieval recipes you might try these:

http://www.celtnet.org.uk/recipes/medieval.html

http://www.medievalcookery.com/recipes/

http://www.godecookery.com/godeboke/godeboke.htm

Medieval Cookery also makes it easy to browse by category or country.

One resource I'm impressed with is Rouxbe.com, which is an "online cooking school" with truly excellent video content. (Disclosure: I'm an affiliate, though I haven't earned anything from it - I only signed up because I think it is great, I don't refer anything just to hope to make a buck). That link will give you a 14 day free trial. The thing I like is that it is broken down into really small, highly detailed sections, and the video is HD and shot extremely close up on the hands and pots so you can really see what is happening.