Which wallets can hold ERC20 Tokens that aren't listed on exchanges?
The hardware wallets are best wallets to store ERC20 tokens safely like Nano Ledger S. If can't buy a hardware wallet, my next option is Metamask. The Metamask is a chrome extension and it has a good security with anti-phishing. Another option is Myetherwallet. Most people using Myetherwallet to store ERC20 tokens. But you have to keep safe your private keys.
Is it possible to send it to my ethereum wallet address on Binance or Poloniex
No, never send your ERC20 tokens to exchanges ethereum wallet addresses. You have to control your private keys to manage ERC20 tokens. The exchanges wallets private keys owned by the exchanges that's why.