Can anyone tell me what our adopted cat Lilly is ?

I am assuming that Lilly is a female (by the name but I have a female named Rocket so I know I could be off). Which means she is a semi-rarity in that only approximately 20% of orange tabbies are females. The code for the orange gene is on the X chromosome. To express as solid orange and as a male cat he only needs his one X chromosome to be coded for orange, but a female has to have both of her X chromosomes coded for orange to be solid orange otherwise she turns out a mix like calico or tortoiseshell. From this we can be 100% sure that Lilly's father was a solid orange tabby cat for her to get one of the two needed genes and her mother was an orange, calico or tortoiseshell to give her the other. 

Orange tabby cats can be called ginger or red or marmalade cats if you like the sound of that better, but those names refer to color pattern not breed. Her stripes aren't very defined right now but I would guess more mackerel pattern to the tabby stripes. Right now the stripes are fainter so they are what we call ghosting or ghost stripes which is a fun term for faint stripes. But as Lilly get older you will likely notice more difference in the striping or there is a chance she may stay more solid looking. Either way still a tabby because all cats have the tabby gene but if they have the two recessive genes that can hide the tabby pattern making them look like a solid color cat or without the tabby patterning; that recessive gene is suppressed by the orange color gene so all orange cats are tabbies.

Lilly's breed is what is called a domestic short hair (DSH) which means that she is of undetermined ancestry and a short hair. Altogether that makes her an orange tabby DSH, although I would likely go with marmalade just for fun.