An example of a response from etherscan API for transactions this:
{"jsonrpc":"2.0","id":1,"result":{
"blockHash":"0xf64a12502afc36db3d29931a2148e5d6ddaa883a2a3c968ca2fb293fa9258c68",
"blockNumber":"0x70839",
"from":"0xc80fb22930b303b55df9b89901889126400add38",
"gas":"0x30d40",
"gasPrice":"0xba43b7400",
"hash":"0x1e2910a262b1008d0616a0beb24c1a491d78771baa54a33e66065e03b1f46bc1",
"input":"0xfc36e15b0000000000000000000000000000000000000000000000000000000000000020000000000000000000000000000000000000000000000000000000000000000a4861636b65726e65777300000000000000000000000000000000000000000000",
"nonce":"0xa7",
"to":"0x03fca6077d38dd99d0ce14ba32078bd2cda72d74",
"transactionIndex":"0x0",
"value":"0x0",
"v":"0x1c",
"r":"0xe7ccdba116aa95ae8d9bdd02f619a0cdfc1f60c5740b3899865822a80cd70218",
"s":"0xf200df1921ea988d16280a0873b69cb782a54e8a596d15e700710c820c8d2a9e"}}
The input is what you are interested in.
The first 4 bytes are the method (function) that is being called.
The remaining are the parameters in words of 32bytes.
If you want to know the function name and types of the parameters you need to know the contract (ABI). Having the ABI you just need to make a quick look for the method.
You can retrieve the contracts you need to know the addresses, there is no way to download all at once. see here
Hope this helps