Oh, you did some mistakes~ "Sarang hamnida" is right. Sarang means 'love', and hamnida means 'do'(respectful expression). So "Sarang hamnida" means '(I) do love (someone)' → 'I love (someone)'

I do agree with Juan Park, however, I must say it would be much easier to treat 사랑합니다 as a single expression. I.e., 사랑합니다 means simply "I love..."

When you love someone, you can say "sa-rang-hae"(사랑해), "sa-rang-han-da"(사랑한다), "sa-rang-ham-ni-da"(사랑합니다.) and "sa-rang-hae-yo"(사랑해요) "I love you." (사랑해 or 사랑한다) In this case, 'you' are youger than me or are the same age of me. I think "sa-rang-hae" is more colloquial than "sa-rang-han-da" If someone hear "sa-rang-hae", he or she feels it more sweet than "sa-rang-han-da". "I love you all." you can say it to audience, This is "저는 여러분 모두를 사랑합니다. (or 사랑해요)" in Korean. "I love you, mom." This is "어머니 사랑합니다. (or 사랑해요)" in Korean.