This is my first notebook entry!

Saying "This is my first notebook" kinda sounds like you bought your first notebook.

Hi, I'm Eriko. I'm a Japanese woman living in England now. I've been studying English at an English school. Learning English is fun and I'm enjoying my school days.
I came to arrived in England in June and was surprised that it was so cool in spite of in it being the beginning of summer. In Japan, I got I was used to having hot and humid summers in Japan so I missed the hot summer in Japan there. But now I'm getting used to the weather here and Having a good time. I'm having a good time here now that I'm used to the weather.

It sounds more natural if "now" was omitted. Saying "arrived in England" sounds better and the two sentences sound more natural when connected with an "and". I think it's more common to use "despite" instead of "in spite of" but both are correct. Using "in Japan" at both the beginning and the end of the sentence feels weird so I paraphrased it a bit. Not sure how to correct the underlined sentence so I paraphrased it instead.

I see people from other countries everyday, particularly Europeans. everyday and I'm thinking of visiting Paris in December so I also want to learn French and but I'm also interested in learning Spanish too.

I'll try to write everyday. I'm writing in English now and but I wish I can could write in French or Spanish someday!

It's more natural to connect the "visiting Paris" part with the "learning French" part. The difference between "and" and "but" is that "and" is like an extension while "but" is like an opposing thing. Not really sure how to explain it any better. For example, "I want to learn French and Spanish" means you want to learn both French and Spanish BUT saying "I want to learn French but I also want to learn Spanish" means you want to learn both but you only have time to learn one. From what you wrote, you want to learn French because you want to visit Paris but you're also interested in Spanish, that's why you should use "but" in this case.

I might have missed some stuff here and there but it should be fine for now.