TIM revised prepaid
 

This suggests that new customers get the first month of internet data for free (to 500mb/week), after which 500mb/week costs 5€
TIMxSMARTPHONE Maxi - Nuovi Clienti | TIM

"The promotion is valid for those who enter into TIM, with number portability, and active at the same TIMxSmartphone MAXI by 03/02/2013"

La promozione è valida per chi entra in TIM, anche con portabilità del numero, e attiva contestualmente TIMxSmartphone MAXI entro il 03/02/2013

Does number portability mean you had to have ported another number to TIM to get the first week free?

To activate SMS to 40916 with the text: TIMSMARTMAXI ON

In addition, TIM prepaid used to charge 2.5€ for 250mb/week. That appears to have changed to 3€ for 300mb/week. To activate SMS to 40916 with the text: TIMSMART3 ON.

TIMxSMARTPHONE | TIM

Am I misreading something?

The first page says the new customer deal is valid to 03/02/2013. The FAQ seems to say sign up by 02/12/2012 " La promozione è valida per tutti i Nuovi Clienti TIM Ricaricabili che entro il 02/12/2012 attivano l'Opzione presso un Negozio TIM, contemporaneamente all'attivazione di una nuova linea TIM o all'ingresso nel mondo TIM da Altro Operatore con portabilità del numero."

What is the deadline?

I think they are 2 different offers. The first one was for all new customers (wether porting their number or not), but it ended on Dec 31st 2012. Now the promotion is reserved to porting numbers only and it will end on Feb 2nd 2013.
In any case TIM should update their website...

If I understand correctly, a SIM card costs 10€ with a 5€ credit and the minimum additional recharge amount is 5€.

I'd want one week of 500mb (which would cost 5€) and a second week at 300mb (which would cost 3€), so that I have some credit available for voice calls.

If so, would I initially pay 15€ for the card plus extra credit, SMS TIMSMARTMAXI ON to 40196 (to activate 500mb), then at the end of the week SMS TIMSMARTMAXI OFF (so that it does not automatically renew) followed by TIMSMART3 ON (to activate 300mb)?

Did I get this right?

Yes, it should work. I think you can send the deactivation SMS also before the end of the week and you should be able to use the option till the end of the week in any case. With Wind it surely works this way, I'm not 100% sure with TIM too, but I hope so...
BTW I checked on TIM website and it seems you have to call 40916 to deactivate the TIMSMARTMAXI, or do it through their website if you registered your card there (which I suggest). It seems the TIMSMARTMAXI OFF command does not work, but it's really strange...

Their web site is not very clear. I also notice they are still advertising the new customer offer that expired 3 February.

It's also not clear if TIMSMARTMAXI renews automatically. The website suggests it doesn't, that they send an SMS and you have to respond "si".

Registering online seems the best, as I don't speak Italian. This means figuring out my phone number, which my prior traveling phone didn't report, but the current one should work.

The special offer is in still in effect. I get a month of data (limited to 500mb/week) upon the purchase of a 10€ SIM card.

Activation was immediate.

I had to show a passport and sign some papers which appeared to authorize a Codice Fiscal.