Well, Coinbase is an exchange, not technically a wallet, but that’s a minor point. Anyhow, there are plenty of crypto currencies you can buy - though the number on Coinbase is fairly limited. There are three basic ways that you can try to make money with crypto currency: trading, interest, and staking.
Trading is basically what you’re taking about with Doge coin, in that you buy a coin for a certain amount of fiat currency (dollars, or whatever is appropriate for your country) and try to sell it for more. You can do this easily enough on Coinbase, with the coins they offer, or sign up for some other exchanges that have the coins that you want to try to trade. Even trading between exchanges can be profitable at times, as prices differences mean that you can buy it for cheaper one place than you can sell it on another. The main thing to watch out for are fees. Exchanges usually charge you a fee to withdraw coins or to buy/sell, plus most cypto currencies have a baked in transaction fee to send the coins from one place to another.
Interest is essentially the same as banks. You deposit crypto, they loan it out and charge interest, and you get paid a small amount of interest while they hold your coins. The main thing to watch out for here is disreputable services and scams. If something sounds too good to be true, it usually is.
Staking is a mechanic that a certain set of coins use to produce new coins. With BTC and Ethereum people need to mine them, that is use computers to do work in order to try to win a reward of new coins being minted - or, Proof Of Work. There is also a set of coins know as Proof of Stake coins: these distributed newly minted coins to holders of the coin. So, if you lock up some coins of this type, they should generate more over time. On Coinbase, Tezos would be the main example of this.
In all cases, you should do research before doing any investing. If you have specific questions, feel free to ask and people on here may chime in with their thoughts/opinions.