My hometown

1/ I probably would not use "small and compact" here. They aren't always the same (Hong Kong, for example is large (a very high population) but compact (the population is crammed into a small area). However when both the population and the area are small, "small" would be enough.2/ You only capitalise proper nouns; the specific names of places. North Dakota, for example. In this case "north" just describes the general region. It's not a place name.  3/ The first letter of a sentence needs to be capitalised.

I have been living in this city since I was born and I can certainly say that there are many interesting things to visit places to visit and things to see.

1/ "I have been living in" is correct, but "I have lived" is a shorter, cleaner way of saying it. 2/ You visit places, but you see things. You can also see places, but you would not normally "visit things".

It is not a tourist place but, actually, there are some charming attractions such as the cathedral and the main square in the city centre.
Furthermore, in the city centre there are some lively bars and restaurants and sometimes there are also some small live concerts in on the weekend.

You always say "on" rather than "in" when referring to a date; "on Saturday", "on the first of May", etc. The only exception is if it falls in a month or year; "I went to Italy in August", the Berlin Wall fell in 1989, etc.

Moving to suburbs, the city is full of parks and facilities for children. Besides, there are a lot of big supermarkets in spite of the city centre where there is a high presence of small local shops.There are a lot of big supermarkets in contrast to the city centre, where there is a high presence of small, local shops.

It's not really "besides" or "in spite of" in this case. You would say "in spite of" (or "despite") if something happened regardless of something that might have prevented it. For example, "The building of the supermarket was approved by the council despite the objections of the small shop holders". In this case two things are different (supermarkets in the suburbs, small shops in the centre) but neither one is "in spite of" the other.

The suburbs are some of the quietest places in of the city because the traffic is less heavy than in the centre.

"Suburbs" is a plural word; they don't describe just one place. You therefore need a plural construction..

In the end, I can state that my hometown is a fascinating place to live in, especially for families. I suggest that visitors go to visit the city museum which has inside many evidence artefacts of the ancient Romans era.

I'm out of space...