There are both hardware, desktop, and mobile wallets that support a large variety of cryptocurrencies.

Hardware - For hardware wallets (the safest multi-currency wallet option), the Ledger Nano S supports the most cryptos including Bitcoin, Ethereum, Ripple, Stellar, Bitcoin Cash, Litecoin, and Tron. In total it supports 39 cryptocurrencies with their own blockchains + all ERC-20 tokens.

Mobile Wallets - There's a variety of options for mobile wallets, but I'd recommend the Edge wallet (available for both Android and iOS). It supports Bitcoin, Ethereum, Bitcoin Cash, Ripple, Monero, Litecoin, Dash, Augur, Zcoin, Feathercoin, DigiByte, Vertcoin, and Bitcoin Gold. It also supports all ERC-20 tokens, whereas many mobile wallets only support select ERC-20 tokens.

Desktop - Jaxx is probably your best bet if looking for a desktop wallet (and it's also available on mobile). You can find a full list of supported cryptocurrencies here. Jaxx however does not fully support all ERC-20 tokens.

You can find a more in-depth comparison of wallets and their supported cryptocurrencies here: https://unhashed.com/best-bitcoin-ethereum-altcoin-wallet-reviews/

The best exchange for you is dependent on your needs as well as some personal preference.

There's currently at least 200 cryptocurrencunhy exchanges, according to https://coinmarketcap.com/exchanges/volume/24-hour/. In general, it's best to stick to the more well-known exchanges (which typically have the highest volume).

Coinbase is often recommended to beginners, as it's the easiest exchange to use and has industry leading security. The major downside of Coinbase is that it has high fees (~1.49% for bank transfer purchases, 3.99% for credit cards (though it's one of a few exchanges to even accept credit cards and these fees are lower than CC accepting competitors)).

GDAX (owned by Coinbase), Gemini, Kraken, and Bitstamp are all exchanges that have fees under 0.4% and allow you to make purchases via bank transfer (Bitstamp also accepts CC purchases). All of these options are widely used and based out of the U.S. or Europe (Bitstamp).

• Gemini supports trading for Bitcoin and Ethereum
• GDAX supports trading for Bitcoin, Ethereum, Litecoin, and Bitcoin Cash
• Bitstamp supports trading for Bitcoin, Ethereum, Litecoin, Bitcoin Cash, and Ripple.
• Kraken supports the above 5 coins and 10+ more.

Once you own cryptocurrency, you trade a huge variety of altcoins on Binance. Binance is the world's most active cryptocurrency exchange and offers over 100 different cryptocurrencies.

Here's a more detailed comparison of the best cryptocurrency exchange: https://unhashed.com/best-bitcoin-cryptocurrency-exchange-reviews/

Bittrex has temporarily closed registration to new users, unfortunately making Poloniex the only option to sign up between these two exchanges at the moment. As others have mentioned, Binance is an other exchange to consider for similar services.

Bittrex vs Poloniex vs Binance Key Points of Comparison

• Customer Support: While none of these exchanges have anywhere close to perfect customer support, Poloniex seems to be particularly bad in this area. Many Poloniex users have experiences support tickets taking multiple months. Binance and Bittrex support is usually handled in a more reasonable time frame of a few weeks max.
• Verification Requirements: Binance by default allows you to withdrawal the equivalent of 2 BTC / day, without requiring any personal information. You can increase this to 100 BTC / day by completing verification. Both Bittrex and Poloniex require verification for any withdrawals. Keep in mind Poloniex's poor customer support may severely delay your verification.
• Current Trading Pairs: Binance 264, Bittrex 261, Poloniex 99. Not only does Binance currently have the most trading pairs, but they've also been most consistent recently in adding news coins to their exchange.
• Trading Fees: Binance 0.1%, Bittrex 0.25%, Poloniex 0.25% or less (maker-taker fee schedule)
  

You can see a more indepth comparison of Bittrex vs Poloniex vs Binance here https://unhashed.com/exchange-comparisons/bittrex-vs-poloniex-binance/

Bittrex has temporarily ceased new signups to allow them to upgrade their customer support and infrastructure. Even when Bittrex reopens signups, Binance will likely still be the better option.

Quick Comparison of Binance vs Bittrex:

• Both sites are cryptocurrency only exchanges, meaning you can trade cryptos for other cryptos. Neither supports fiat currency deposits, withdrawals, or trading.
• Bittrex requires at least a name, address, and date of birth to withdrawal. This info only allows you to withdraw 0.4 BTC / day. Binance only requires an email to be able to withdraw the equivalent of 2 BTC / day.
• Both sites offer a huge variety of altcoins to trade, though Binance has been more consistent recently with adding new coins.
• Binance is the world's most active cryptocurrency exchange in terms of daily trade volume. Bittrex usually is a few ranks behind Binance.
• Binance offers a mobile app, which Bittrex does not.
• While neither exchange's customer support is perfect, Binance is likely a bit better.

You can see a more detailed comparison here: https://unhashed.com/exchange-comparisons/bittrex-vs-poloniex-binance/

It's likely due to multiple factors, including the South Korean news, other negative news stories, and recent surges in altcoins.