Hi there, here are my corrections.  I hope this is useful, let me know if you have any questions about them!

We moved into new place / house to live yesterday after finishing our stay at a hotel.
As I wrote it down before, the stay at the hotel was so tiring. Finally I have got to a place where I can be relaxed.

Today I went to the shopping centre near my house. Before getting there, I dropped by the beach. When I was sitting on the a ('a' and not 'the' because this is the first time we have heard about the bench)bench, an old woman sat next to me and spoke to me. She said she is was 83 years old, and lived by herself enjoying gardening and spiritual things. Actually she looked healthy but at same time she looked lonely. And Also, (we don;t usually start a sentence with 'and')I couldn't speak English well and couldn't lesson understand her English well.

Coming When I came ('coming back home' is a classic case of using the 'ing' form also to mean 'when'  The 'ing' form sounds best when one event interrupts another event.  e.g. 'Walking very quickly, I fell over and hurt my leg' - Walking was interrupted by the falling over.  In your example, one event simply starts only AFTER the other event is finished)back to home, I made pasta and some side dishes for dinner. I'm sure that it's better to have dinner at home than going to go (infinitive is needed here because you used infinitive with 'to have dinner')out.

That's it. Good night!

Diary

We moved in new place to live yesterday after finishing our stay at a hotel.
As I wrote it down before, the stay at the hotel was so tiring. Finally I got have a place where I can be relaxed.

Today I went to the shopping centre near my house. Before getting there, On my way,* I dropped by the beach. When I was sitting on the a bench, an old woman sat next to me and spoke began talking** to me. She said she is was*** 83 years old, and lives**** by herself enjoying gardening and spiritual things. Actually she looked healthy but at same time she looked lonely. And I couldn't speak English well and couldn't lesson had a hard time understanding her English well.

Coming bBack to at home, I made pasta and some a couple of side dishes for dinner. I'm sure***** that having dinner at home is it's better to have dinner at home than going out******.

That's it. Good night!

I'm happy to hear you are finally in your new home!

* "On my way. . ." - This means almost the same thing as "before getting there" but it emphasizes that you were already on your way to the shopping center (US spelling - sorry if it's confusing) and decided to drop by the beach.

** The difference between "speaking" and "talking" is very subtle. "Speaking" means the skill of being able to speak. "Talking" usually means having a conversation. Often, both words can be used interchangeably.

*** "She was" - She still is 83, but because you are telling the story in Past Tense, you should keep the verbs in Past Tense.

**** I think this is correct. Even though it contradicts what I just wrote in the above note, it sounds very natural to say "lives by herself"

***** "I'm sure that . . ." is okay. But I think saying "I feel that . . ." sounds more natural. The meaning is very similar to と思う or と感じています.

****** Also, like David said in his correction, the verb tenses need to "agree" with each other. But I think that the gerund form ("having dinner at home . . . going out") sounds better than the infinitive ("to have dinner at home . . . to go out"). A gerund is a noun-form of a verb. The "~ing" ending is similar to adding "~こと" to a Japanese verb.