Mining Ethereum with AMD GPUs
Ethereum is considered by WhatToMine.com to be the benchmark alt coin. It is not always the most profitable cryptocurrency to mine, but with its enormous market cap volume between 25 and 30 billion US Dollars, it is about as steady and stable as alt coins come. It is also probably the most widely recognized and accepted cryptocurrency that isn't Bitcoin. The Ethereum World Computer project brings the potential for numerous digital products and services to be exchanged and even run from the peer-to-peer cloud, all fueled by Ethereum. This decentralized network of currency, products, services and computing/serving power has hardly been tapped yet, as hardly anyone is aware of it. These factors make Ethereum a very interesting crypto coin to experiment with.
Ethash is one of the algorithms that AMD cards seem to have an advantage in mining. The RX 470, 480, 570 and 580 are some of the most cost-effective GPUs for mining Ethereum and other Ethash cryptocurrencies. Vega cards are too new and overpriced at this point to tell where they will fall on the ROI scale, but they have been tested at this early stage in their release and reportedly achieve 35+ Mh/s at stock settings; very impressive. The low cost of other AMD based cards and their ability to mine Ethash coins like Ethereum at nearly the same hashrates as more expensive Nvidia cards would suggest that it is wisest to invest in the last two generations of AMD GPUs if you want to specialize in Ethereum and other Ethash cryptocoins, particularly if you are willing to flash the BIOS or pay someone to do it for you. It is a risky proposition, so I cannot recommend anyone who isn't experienced with modifying firmwares attempt it themselves, but I do have an explainer video on the main AMD Mining page.
Claymore Dual Miner
You can find the latest releases for Claymore for Windows and Linux on Nanopool's Github. Searching numerous crypto-mining message boards showed that Claymore is presently the fastest miner to use for mining Ethereum in Windows 10. It works for both Nvidia and AMD cards. It is also capable of dual mining an Ethash coin with another coin on a different algorithm without sacrificing much hashrate on the Ethash mining. The secondary coin usually gets a very small amount of performance as seemingly 90% or more of the hashing power goes to the main coin. I will try to make a page soon specifically about dual mining to avoid this page becoming too long... (DONE! See here: AMD Dual Mining)
Ethereum Wallet and Mist
Before you start mining, you will need a wallet to mine your Ethereum into. There are a few choices, but the official Ethereum Wallet coupled to Mist is on the Ethereum Github Ethereum has been around for a long time and has built quite a large blockchain over the years. In order to sync your node, you are most likely going to want a modern computer, preferably with a solid state hard drive to minimize the time it takes to catch up with the blockchain starting from scratch. Even then, it probably is not going to happen for you in less than a day.
I am a little short on time at the moment or I would go deeper into getting your wallet installed and sync'd up. I will try to get that added soon, but it is not difficult and the instructions on GitHub should suffice for sufficiently advanced users.