SIM cards for cell phone

S.L.,

One important point to mention is that in order to use any other SIM cards in your phone, the handset must be unlocked.

I don't know where the cell shops in Venice are located. I'm sure the staff at your hotel will be able to direct you to the nearest one. Yes, the cards will work in other countries but the cost will be slightly higher since you'll be "roaming". In some cases, it may be difficult to top-up the cards once you've left the country of purchase.

You can buy Vodafone Holiday or TIM Welcome. Those are packages designed for foreign tourists.
Once you leave Italy they work in Spain or France. However you will be roaming. The best way to avoid huge roaming charges and deplete your SIM credit in one hour, is to activate a Smart Passport plan which (for Vodafone for example) costs 3 euro a day. To do so you need to add money to your plan and activate it using the phone or online. Vodafone for example (I'm a vodafone customer) will deduct 3 euro automatically from your credit available in your phone every day (if you use the phone, otherwise no deduction). So the day before leaving, go to a telephone store, load additional money on the plan and ask the assistant to activate the plan (the voice prompt might be in Italian only). Everyday outside of Italy the provider (Vodafone) will take 3 euro from your credit. But you can use the plan while in other countries (including the US). Once the credit is gone, they stop deducting (or you can stop the Smart Passport plan over the phone, but I think it's only italian voice prompts), so put enough to cover you for your stay after Italy. You don't say how long you'll stay outside of Italy (France/Spain). Obviously if you plan to stay 2 weeks in Spain after Italy, then it might be cheaper to buy another Spanish SIM, rather than spending 3 euro a day on the Vodafone Smart Passport. But for example if you stay only 5 or 6 days in Spain and France, then the additional 15 or 18 euro on the smart passport might be the cheaper option.

Just returned. There is no place to get a sim card in Venice airport. You can get one in Venice, I used TIM chip It gave a couple of gigs for data - don't remember how many but their package they offer to travellers only gives $5 of voice which can be depleted quickly. You can probably add more voice time as needed but I didn't try... I mainly needed data

Vodafone Holiday gives you 300 minutes (anywhere), 300 SMS texts, 2 Gb for 30 euro.
http://www.vodafone.it/portal/Privati/Tariffe-e-Prodotti/Tariffe/Estero/Vodafone-Holiday-English.
If you intend to use it when you also go to Spain, in order to buy the Smart Passport option for roaming abroad (3 euro/day) you need to top up the credit on the card and add 3 euro for each day you intend to use it outside of Italy.

TIM Welcome offers 200 minutes (anywhere) and 2Gb for 20euro. However it requires also an additional 10 euro for the purchase of the SIM card which comes with a 5 euro credit. Since it doesn't mention any SMS text messages included, I guess you can use the 5 Euro credit for the text messages.
https://www.tim.it/offerte/mobile/estero/dallitalia/tim-welcome-it

I think Vodafone is a better deal (don't tell my cousin who's a manager for TIM). I use Vodafone also because I have poor reception with other providers at my country house in Tuscany.

There is a Vodafone Store in Venice next to the Rialto Bridge.
http://trovanegozio.vodafone.it/venezia/autoservice-2-srl/
or at Cannaregio, not far away from the one above.
http://trovanegozio.vodafone.it/venezia/aska-srl/