Corrections in red; stylistics in blue.

Present simple - past simple sentences

I like to listen to live instrumental and good vocal music.
I also like a healthy lifestyle, in particular, various physical exercises: running and swimming. Sometimes I read books and watch movies. On the weekends, if the weather is good, my friends, relatives and I [1] go out to relax in nature or visit the sauna. I like to eat fruit and vegetables, as well as a variety of meats.

I liked to listened to live instrumental and good vocal music good.
I also liked a healthy lifestyle, in particular, various physical exercises: running and swimming.   [2]. Sometimes I read books and watched movies. On the weekends, if the weather was good, my friends, relatives and I would go [3] out to rest in nature or visited [3] the sauna. I liked to eat fruit and vegetables, as well as a variety of meats.

[1]  In Russian you can use the formula "Мы с (кем-то) + глагол."  You cannot use this formula in English.  Instead you say, "(Someone) and I + verb".  Example:  "Мы с Иваном друзя". = "He and I are friends."  

[2]  "running" and "swimming" are gerunds, which function like nouns.  Verbs have tenses.  Gerunds do not have tenses.
 [3]  When a action was performed from time to time in the past, it is often appropriate to use the word "would" plus the present tense form of the main verb.  The past simple tense does not sound right.  Example:  Я писал каждый день. =  I would write every day. [Variant:  I used to write every day.]