I would personally phrase it in the following 2 ways:

１：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}について話{はな}していきたいと思{お}います。- I would like to talk about the place I like in this "essay".

or

２：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}を紹{しょう}介{かい}していきたいと思{お}います。- I would like to introduce (to the reader) the place I like in this "essay".

I think both are fairly used. I will try to breakdown one of the sentences so you understand how all the bits work.

この作文の中で - In this essay - Pretty straight forward.

(私の)好きな場所 - The place I like - It is ok to omit the "I" in Japanese.

について - About - Nothing to add here.

話していきたい - I want to talk / I want to go and talk - it has the nuance of going to(していく). It is the たい-form of 話していく（していく).

と思います - I would like to - したいと思います is polite way to say that you want to / will do something.

Now for the second sentence, the only difference is we are changing 話す for 紹介する, which means to "introduce" and it is used a lot in those situations where you are talking about something you know, but your audience does not. In that case we will just change について話す for を紹介する (していきたいと思います).

I would personally phrase it in the following 2 ways:

１：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}について話{はな}していきたいと思{おも}います。- I would like to talk about the place I like in this "essay".

or

２：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}を紹{しょう}介{かい}していきたいと思{おも}います。- I would like to introduce (to the reader) the place I like in this "essay".

I think both are fairly used. I will try to breakdown one of the sentences so you understand how all the bits work.

この作文の中で - In this essay - Pretty straight forward.

(私の)好きな場所 - The place I like - It is ok to omit the "I" in Japanese.

について - About - Nothing to add here.

話していきたい - I want to talk / I want to go and talk - it has the nuance of going to(していく). It is the たい-form of 話していく（していく).

と思います - I would like to - したいと思います is polite way to say that you want to / will do something.

Now for the second sentence, the only difference is we are changing 話す for 紹介する, which means to "introduce" and it is used a lot in those situations where you are talking about something you know, but your audience does not. In that case we will just change について話す for を紹介する (していきたいと思います).

I would personally phrase it in the following 2 ways:

１：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}について話{はな}していきたいと思{お}います。- I would like to talk about the place I like in this "essay".

or

２：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}を紹{しょう}介{かい}していきたいと思{お}います。- I would like to introduce (to the reader) the place I like in this "essay".

I think both are fairly used. I will try to breakdown one of the sentences so you understand how all the bits work.

この作文で - In this essay - Pretty straight forward.

(私の)好きな場所 - The place I like - It is ok to omit the "I" in Japanese.

について - About - Nothing to add here.

話していきたい - I want to talk / I want to go and talk - it has the nuance of going to(していく). It is the たい-form of 話していく（していく).

と思います - I would like to - したいと思います is polite way to say that you want to / will do something.

Now for the second sentence, the only difference is we are changing 話す for 紹介する, which means to "introduce" and it is used a lot in those situations where you are talking about something you know, but your audience does not. In that case we will just change について話す for を紹介する (していきたいと思います).

I would personally phrase it in the following 2 ways:

１：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}について話{はな}していきたいと思{お}います。- I would like to talk about the place I like in this "essay".

or

２：この作{さく}文{ぶん}の中{なか}で、（私{わたし}の）好き{すき}な場{ば}所{しょ}を紹{しょう}介{かい}していきたいと思{お}います。- I would like to introduce (to the reader) the place I like in this "essay".

I think both are fairly used. I will try to breakdown one of the sentences so you understand how all the bits work.

この作文の中で - In this essay - Pretty straight forward.

(私の)好きな場所 - The place I like - It is ok to omit the "I" in Japanese.

について - About - Nothing to add here.

話していきたい - I want to talk / I want to go and talk - it has the nuance of going to(していく). It is the たい-form of 話していく（していく).

と思います - I would like to - したいと思います is polite way to say that you want to / will do something.

Now for the second sentence, the only difference is we are changing 話す for 紹介する, which means to "introduce" and it is used a lot in those situations where you are talking about something you know, but your audience does not. In that case we will just change について話す for を紹介する (していきたいと思います).