It's not that hard... Раз, два, три, четыре, пять, шесть, семь, восемь, девять, десять. When it gets tricky is when you want to count a specifc object. Одна девочка, две девочки, три девочки, четыре девочки, пять девочек, шесть девочек, семь девочек, восемь девочек, девять девочек, десять девочек.
Basically it can be summed up as, in nominative, "one X" (and compound numbers ending in 1) uses the nominative singular form of X and "one" agrees in gender. Two, three, and four (and compound numbers ending in 2, 3, or 4) X use the genitive singular form of X. Two (and compound numbers ending in 2) also agrees in gender. Everything else uses the genitive plural form of X.
Simple, right? ;)
When you use numbers in Russian, the case of the counted noun depends on the last digit of the number. There's a detailed breakdown here on RussianLessons.net, but basically:
For numbers ending in 1--but excluding 11--the noun takes the singular version whatever case it would otherwise have. (один will also decline like an adjective. двадцать одна кошка видели меня, "Twenty-one cats saw me", but Я видел двадцать одну кошку, "I saw twenty-one cats.")
For numbers ending in 2, 3, or 4 (except for 12, 13, and 14), the noun takes the genitive singular, regardless of its role in the sentence.
For numbers ending 5, 6, 7, 8, 9, 0, and all -надцать numbers (11-19), the noun takes the genitive plural, regardless of its role in the sentence. The "-надцать" rule takes precedence over all others; whether it's 11 or 111 or 1011 or 982350982711, they're all considered to be ending in 11 (одиннадцать), which means they're all -надцать numbers and they all make the nouns they're counting take the genitive plural.
I very strongly recommend you to use wiktionary, it has been an invaluable tool in my learning so far. You can check all forms of кошка here:
There you can see that, one is genitive plural, and the other nominative plural.