I get $348.00 pension from my late husband. I get this amt in one year. I am on SSI for the rest of my income. Do I need to file taxes on this money?

SSI payments are not income.
With $348 of pension income, you do not need to file tax return. For single status the filing requirement starts with $9450.

Agreed; your income level is WAY below the minimum required to file.

I'm on diabilty. How much can I make without paying taxes,before taking out my deductions.I work about 15 hrs. a week.

Depends on the nature of your disability; some disability payments are tax-exempt; some are not. Further, disability in some cases becomes taxable if added to other income.

Also, we would need to know HOW MUCH you make per week.