'그리고' is only used at the beginning of a sentence to mean "And..." When you're linking two sentences with "and" (in the sense of a simple sequence but not implying any causality between the first and second sentence), you can also use '-고' on the end of the verb stem of the first sentence.
As for '와', I assume you're referring to the pair '-와' (attached to nouns ending in a vowel) and '-과' (attached to nouns ending in a consonant). These are used to link two nouns (one of the two can be implied) in the sense of 'and' or 'with'.
Some examples:
1. I had Chinese food. And I had dessert, too. -> 나는 중국음식을 먹었어요. 그리고 디저트도 먹었어요.
2. I had Korean food and I went home. -> 나는 한국음식을 먹고 집에 왔다.
3. I like apples and peaches. -> 나는 사과와 복숭아를 좋아해요. (see the '-와' linking '사과' and '복숭아')
4. I went to the market (together) with my mom. -> 엄마와 같이 시장에 갔어요. (here you're adding adding '-와' to '엄마' but instead of Noun 1 + 와/과 Noun 2, you're implying one of them. It's like saying 엄마와 나 (mom and I))