How do I get old texts onto new iPhone?

You might just take it into an Apple Store and see if the Genius Bar person will just grab the contacts-transfer tool they have for new phones and move your contacts for you. They used that thing to move my wife's contacts from her old SIM to a her new phone in about 15 seconds. It also had connections for basically every other phone, and according to the guy, it worked with varying degrees of success depending on the phone.

Rockindata, do you think I should activate the Micro SIM before or after attempting this?

Verizon has a backup tool called Backup Assistant that will save all your contacts over the air to their server. You can go online to MyVerizon and access them, download as a CSV, and then transfer them to your iPhone (probably the easiest is to import them into what ever email account you will be using/syncing on your iPhone). I have never done this at the Apple store, but if you go that route you will probably want to activate the phone first. I would try Apple last though, because once you disconnect the Verizon phone from service you won't be able to do anything over-the-air with it - although you can still access your contacts and all that, you just can't make phone calls or send messages or do backups. Not sure about pictures. Does your phone have a slot for a mini- or micro-USB card? What about connecting the phone directly to your computer? You might be able to go to a Verizon store and get help (just tell them you want to back up your pictures, no need to tell them you're leaving for another provider). You can also email them to yourself via text message (just put the email address in the phone number field) - but if you don't have an allowance for picture messages this will get expensive, fast. You're SOL with texts.

Thanks everyone. I ended up taking both phones to the Apple store (no SIM in the iPhone) and they were able to move the contacts, but that's it. I backed up the photos onto Verizon's "Online Album" and saved them from there onto my hard drive. As for the texts, I was able to forward them to my email address (typing email address into phone number field). For anyone else switching from Verizon to Straight Talk: porting the number took several hours; I only realized it was done when I tried to send a text and got "message failed." Data only started working when I went to www.unlockit.co.nz to Create APN (through Safari with the iPhone on wifi). Still need to get the picture messaging working.