Today

1/ "Today is Saturday" and "I have to go to the laundry" are two separate thoughts. They therefore need either some form of conjunction (like "and") or they need to be made into two separate sentences.

2/ English uses the article (the / a / an, equivalent to il / un etc) less than Italian but in this case you need one. You say "the laundry" (rather than "a laundry") because you are obviously referring to a specific one, even though you don't name it.

3/ Words that end in "ly" are normally adverbs; they modify a verb. You would use "usually" if you were, for example, modifying the verb "to go" such as "I usually go to the laundry on Saturday" / "Di solito, vado... etc". However in this case "as usual" is a noun (sostantivo); "as usual" means that it is your habit (abitudine) to do this on Saturday.

4/ You don't need to repeat the fact that it's Saturday; that was already covered at the start of the sentence.

The laundry is not far from my home, and it is a laundry which is not expensive for my pockets.

1/ When you say that the laundry is "not expensive" you need some form of "to be" (essere). In this case I have suggested "which is". What you have written there is like writing "La lavanderia non costosa" invece "La lavanderia non è costosa".

2/ "Not expensive for my pockets" is not something that you would hear in English. I understand what you're getting at and certainly using a metaphor like "The cost isn't hard on my pocket" is OK, though I think you probably need to concentrate on getting a really good understanding of literal language first.

When a person lives alone or lives away from home for work, having  services like supermarkets and laundries nearby is remarkably useful and time saving.have near own home comfort as supermarket,laundry and Others services that they are useful for save a lot of time it is a thing remarkable!

1/ The singular ("a person") uses the form lives. The plural ("people") uses the form live. Thus, "a person lives", while "people live". 2/ Supermarkets, laundries and so on aren't normally called "comforts"; they are often called "services", sometimes "facilities". 3/ A service can be useful to save a lot of time,  or for saving time. It would never be "for save a lot of time".

It was a good effort; by all means keep practising.