После прогулки по парку мне самой хочется лечь пораньше, но сначала я мою бутылочку и убираю игрушки.

Because после requires the genitive case.

• Dictionary form: прогулка
• Genitive singular: прогулки

So после прогулки means after the walk or after a walk.

This is a very common pattern:

• после работы = after work
• после ужина = after dinner
• после фильма = after the film

Here по means something like around / through / along when talking about movement within a place.

With this meaning, по usually takes the dative case.

• Dictionary form: парк
• Dative singular: парку

So прогулка по парку means a walk in/around the park.

This is a common pattern:

• гулять по городу = to walk around the city
• ходить по магазину = to walk around the store

Мне is the dative form of я, and самой adds emphasis.

Together, мне самой means something like:

• I myself
• even I
• I personally

In this sentence, it gives emphasis to the speaker’s own feeling: I myself feel like lying down early.

Because the speaker is feminine here, it is самой. A masculine speaker would say мне самому.

Russian often uses хочется as an impersonal way to express a feeling of wanting.

• я хочу = I want
• мне хочется = I feel like / I have the urge to / I kind of want to

So мне хочется sounds a bit softer and more natural for a passing feeling or desire.

Grammatically:

• мне is in the dative because it marks the person experiencing the feeling
• хочется is a fixed impersonal form

Лечь is the perfective verb, and ложиться is its imperfective partner.

Here лечь is natural because the speaker wants to do a single completed action: lie down / go to bed earlier.

• лечь = to lie down, to go to bed once
• ложиться = to be lying down / to go to bed regularly or as a process

So:

• хочется лечь пораньше = I feel like going to bed a bit earlier

Also, learners often confuse this with ложить. In standard Russian, ложить by itself is generally not the standard choice here. The usual intransitive verb is ложиться, and the perfective form is лечь.

Пораньше means a bit earlier or somewhat earlier.

It comes from раньше = earlier, and the prefix по- often makes it sound softer or less absolute.

So:

• лечь раньше = to go to bed earlier
• лечь пораньше = to go to bed a little earlier / fairly early

It often sounds very natural in everyday speech.

Here мою is the verb I wash, from мыть.

• я мою = I wash

So:

• я мою бутылочку = I wash the little bottle / baby bottle

It happens to look exactly like мою meaning my in the feminine accusative, but the sentence structure makes it clear that this is the verb:

• сначала я мою бутылочку = first I wash the bottle

Бутылочку is a diminutive form of бутылка.

• бутылка = bottle
• бутылочка = little bottle

Russian uses diminutives very often, especially for:

• children’s things
• everyday household objects
• affectionate or gentle tone

In this sentence, бутылочку very likely refers to a baby bottle or simply gives a caring, domestic tone.

Grammatically, it is accusative singular because it is the direct object of мою.

Because игрушки is the accusative plural, and for inanimate nouns, the accusative plural is usually the same as the nominative plural.

• Nominative plural: игрушки
• Accusative plural: игрушки

So:

• убираю игрушки = I put away / tidy up the toys

If the noun were animate, the accusative plural would often look like the genitive plural instead.

Also, убирать игрушки usually means to tidy up the toys or put the toys away, not literally just remove them.

Because the two clauses use different grammatical structures.

First clause:

• мне самой хочется лечь пораньше
• This is an impersonal construction
• The experiencer is in the dative: мне

Second clause:

• но сначала я мою бутылочку и убираю игрушки
• This is a normal sentence with a subject in the nominative
• So it uses я

So the switch is completely normal:

• мне хочется = I feel like
• я мою, я убираю = I wash, I tidy up

Russian word order is flexible, but this order is very natural.

• но = but
• сначала = first
• я мою бутылочку и убираю игрушки = I wash the bottle and put away the toys

Placing сначала near the beginning emphasizes the sequence:

1. first I do these chores
2. only after that can I lie down

You could change the word order for emphasis, but the given version sounds neutral and natural in everyday Russian.