I will prefix this answer with the statement "Don't use a web wallet", but since your question explicitly states you want to use one, i'll leave my ideals at the door.
(Side note: Coinbase lets you use their "vault" in multisig mode, where you control the private keys yourself. Withdrawals take 48 hours to confirm, so perhaps it isn't ideal for you, but consider solutions where you retain control of your keys. I am in no way affiliated with Coinbase, other than using their service myself.)
Usually most wallets have an "Export private keys" feature, and an "import" feature included. Which wallets in particular are you using?
This question indicates the following steps for Bitcoin QT:
To export a private key from your Satoshi bitcoin-qt client:
- launch your bitcoin client as usual and wait for it to load the blockchain and start up
- click on 'help' in the menu bar (top right)
- click on 'debug window'
- select the 'console' tab
- type: walletpassphrase "your walletpassphrase here" 600
- type: dumpprivkey [your public key here]
- this will return the private key, you can copy it now; ensure you clear your clipboard/history afterwards
- type: walletlock
For MultiBit, This page describes the steps to export:
In the 'Wallets' side panel choose the wallet you want to export the private keys for.
Select the menu option 'Tools | Export Private Keys'. The 'Export Private Keys' screen appears.
By default, the private keys file is called the same as your wallet file, only with an extension 'key' rather than wallet. You can see this in the following screenshot: [redacted]
If you want to output the private key file to a different directory or filename click on the 'Export to. . .' button. A file chooser opens and you can select the file to save to.
It is strongly recommended that you password protect your private key export files. To do this, click on the 'Password protect export file' radio button.
Enter the password you want to use in the 'Password' field and then repeat the same password in the 'Repeat password' field. If they match you get a little green tick on the right hand side.
To output the private keys to the output file specified, click on the button 'Export private keys'. The private key file is written and a success message appears. If the export of the private key file fails for any reason the error message is given on screen.
After the export file is written, it is then verified by reading the file back in again and checking the keys in the file match the keys in the wallet. A verification message is given if this is successful.
If you can indicate which wallet you're using, I can add the steps for exporting to this post. Additionally, if you want to import into a new service, you will first need to determine whether your wallets are hierarchical deterministic (HD). If so, the "private keys" are in reality a single key, which is used (in effect) as a seed for the entire set of keys in your wallet (theoretically infinite). If your web wallet supports HD key imports, you're all set, otherwise you're going to need to figure out how to extract the private keys for each public key you used, and export those.