"Ten dzień to nowy dzień."
Translation:This day is a new day.
'Today is a new day' = 'Dzisiaj jest nowy dzień' Meaning is very similar but in 'Ten dzień to nowy dzień', 'dzień' is more emphasized.
When you want to say for example 'It/This/That is my dog' you can say 'To jest mój pies'. You can also omit the verb and say 'To mój pies'. Both sound equally natural. However, there are contexts where this contruction doesn't sound natural without a verb (or even is incorrect) so you should be careful using it.