Oh I'd go better than the E10K if possible. That's pretty much lower end entry level and the HD 600 much like the HD 650, scales quite well with better equipment.
You could go with a computer soundcard with a headphone amp. That's where I started with an Asus Xonar STX ( discontinued and replaced with the STX II ). This is a good option if you also plan to play games and need virtual surround modes. Prices range from about $50 up to around $250 depending on the amp/DAC and features the card has.
Entry level discreet devices would include the above mentioned Fiio E10K and the Schiit Fulla 2. Both are amps with included DACs.
Moving up to what I would consider 'audiophile' entry level would be where I'd recommend starting for a headphone as good as the HD 600s. There are generally two products recommended here. A Schiit stack consisting of the Magni 3 amp and the Modi 2 DAC and the Objective 2 amp paired with the Objective DAC ( O2/ODAC Rev B ). The latter is made by both Mayflower Electronics and JDS labs here in the US. The O2/ODAC is known for its complete neutrality while the Schiit stack has more power. You can't go wrong with either. I have a Mayflower O2/ODAC Rev B.
While those are the most commonly recommended entry level amps/DACs I'd also include the Aune X1S ( has both an amp and a very good DAC ) in there but I'd only buy it when it's available from Massdrop. Massdrop will offer it for $200 while it's more than $300 buying it at say Amazon.
Then we can get to more expensive stuff. But there is so much I'm not even going to start listing it. I will say that I have a Topping D30 DAC paired with a class A Aune X7S amp and the combo is amazing with all my headphones but the HD 600s and HD 650s in particular. If bought from Massdrop ( not always available ) the two can be had for right at $300.
Don't even worry about it yet but the HD 600's 300ohm impedance makes it very suitable for OTL tube amps. This isn't something that should be a first purchase though.
Headphones are the most important part of the chain. The amp is next. A good DAC is transparent and while there can be slight sound differences it is the least important part and I'd bet that once you get to the ODAC/Magni 2 level very few people could tell a difference in blind listening.
I love my 600s. If they broke I'd replace them in a second. I also love my HD 650s but if my HD 650s broke I might try something different.
Here's a list of products with links to them and links to their reviews. Zeos, who does the list is quite colorful and his video reviews are usually NSFW ( language ) but he's reviewed just about every good headphone and amp/DAC on the market.