Can I use my old TMobile phone with Nextel/sprint Service?

No,You can use your T-Mobile phone with Nextel/sprint service.Because T-Mobile is GSM,Nextel/sprint service is CDMA.
But you will be able to use other GSM carriers.If you are interested to use other GSM carrier on your mobile phone means,you will need to unlock your phone.For unlocking you can approach your service provider or else try to buy an unlocking service from any one of the online unlocking service provider like Unlock-Zone.com Here you can unlock your phone at low cost.