I want to get 3 backyard chickens for pets & eggs- Please Help

You can definitely keep different breeds together. All breeds usually get along - some people would recommend not mixing breeds with huge size differences and maybe trying not to keep extremely docile breeds with breeds that are more high strung, but I haven't had a problem with breed combinations yet. As for the best combo, just pick what you'd like best! If you want a lot of different colored eggs, maybe one easter egger, one breed that lays brown eggs, and one breed that lays white eggs. If you like different features, like feathered legs and crests, you could pick breeds that have those traits. Really it just depends on what you personally would like from your flock

 

Though some books suggest that birds of the same breed will gravitate toward each other, you can totally mix and match whatever breeds you want. My only suggestion would be to pick birds of similar size (no bantams and large fowl, though this can work). A few good layers are: Rhode Islands, Leghorns, Welsummer, and Sex-Links. Breeds known for really-sweet temperaments are usually not the greatest layers, and tend to go broody. Other than that the temperament really depends on the individual bird.

 

For best breed to identify the sex, you want sex links, hands down. They're not a real breed, but they are mixed breeds bred specifically so you can tell their sex at hatch by their color. A sex link would be fine with your other breeds of choice, and they come in different colors, too, so you could pick what you like best.

 

I would also consider the size of your chook house when deciding on your chooks. My original chook house is quite small. I had australorps before, and now I have isa browns. Being bigger, I did feel that the australorps needed more room than my "little" chooks.
If your chicken house is compact, consider getting "little" chooks and not the "whoppers".