For summer dresses I like a soft cotton lawn as a lining, it's a nice silky feel next to the skin, aids wicking away perspiration, and doesn't add bulk. For silks I like silk as a lining, or one of the new polyesters that shares many of the properties of silk including wicking away perspiration. Taffetas are nice to use as long as they aren't the heavyweights.
As for an online site it does depend where you are as all my fabric sites are in England .
One of THE best sites for fabric is
THe guy is fair, generous and if you're not happy with the fabric, he includes a postage paid label! I have bought TONS of stuff from him and have never, ever been unhappy!