Consonants Zhi, Chi, Shi, Ri, Zi, Ci, Si
Although these consonants do not occur in English, native speakers of English rarely have trouble with them. The most efficient way to start with these consonants is to make sure you can pronounce these few examples:

One “Riben ren shi bu shi chi zhi” [This rather nonsensical sentence means “Don’t Japanese people eat paper?”]

Two “Zheige zi xie si ci” [Write this character four times.]

Three.
Didi xihuan ni. [Younger brother likes you.]

Note that in “Didi xihuan ni,” words with the letter “i” all rhyme with “ni,” not with zhi chi shi etc:

The very short explanation is that Pinyin is not 100% phonetic, and the “letter” i has two pronunciations. After zh, ch, sh, r, z, c, and s, “i” does not rhyme with “Ni, didi, xi, ji,” etc. You could shorten the rule to omit the “h” and say “after z, c, s, and r, the letter i does not rhyme with ni, didi, xi, ji, etc.” To see more explanations of pronunciation rules, please refer to the “pronunciation basics” tab. For now, just know that if you get familiar with the common vocabulary for the z(h), c(h), s(h), and r words in the Quizlet and further below. That will solve most of your needs for these cononants.

Common zhi chi shi words as text
Many finals combine with zhi chi shi. But if you take the commonly used words taught in the typcial textbooks used in freshman level Chinese (university) you will find that many Mandarin syllables are not encountered. Here is a selection as text, all covered in the above Quizlet link.

zhi chi shi
zhīdao [to know] 知道
chīfàn [to eat] 吃饭
shì [is] 是

zh ch sh + a
chá tea 茶

zh ch sh + e
zhè this 这
chē vehicle 车
shé snake 蛇

zh ch sh + u
zhū pig 猪
chūqu go out 出去
shū book 书

zh ch sh + ao
zhǎo look for 找
chǎo stir fry 炒
shǎo few 少

zh ch sh + ou
zhōumò weekend 周末
chòudòufu stinky bean curd 臭豆腐
shǒu hand 手

zh ch sh + uo
zhuōzi table 桌子
shuō speak 说

zh ch sh + an
chēzhàn bus stop 车站

zh ch sh + en
zhēnde ma? Really? 真得吗?
Chén last name 陈

zh ch sh + ang
zhāng measure word; three pieces of paper 三张纸
chàngge sing 唱歌
shàng up, last 上

zh ch sh + eng
zhèng “ing” : zhèng chàngge singing 正唱歌
Zhōngguóchéng Chinatown 中国城
shēngrì birthday 生日

zh ch sh + ong
Zhōngguó China 中国

zh ch sh + uan
zhuānyè major in school 专业
Sìchuān-Sichuan 四川
chuānyīfu wear clothes 穿衣服

zh ch sh + un (wen)
Zhúnbèi prepare to 准备
Chūntiān Spring 春天