That router does not have a adsl port in it. It is unlikely you running ADSL2+ anyway, that protocol is limited to 16m or 32m I forget which.
I suspect you are on a optical solution that uses a specialized modem. Many times the ISP takes care of this and delivers ethernet so you do not need to add another modem.
In any case you are better off with a modem and router separate it give you much more selection. Even with the best router and modem you will get nowhere near the speed they are advertising, it is one of the numbers you can only get in a special test case.
I am in Tallinn, Estonia - they provide the speeds of 100mb/20mb. If you're not getting those speeds you can complain. This is not England (my home country) where companies can rip you off with false advertising.
I just want a unit that I can plug my ethernet cable straight into, and it will work with maximum speeds. I have a router/modem all-in-one that does that now, but it's just old and slow.
Will the Netgear N600 3700 be sufficient to give me this maximum speed? Or do I NEED a seperate modem to deal with the high speed?
You need a modem someplace I suspect. It depends what is actually coming out the wall. If it is normal ethernet then you just need the router and the ISP has handled the modem. You need to ask the ISP this question. If they don't understand ask them if it will work to directly plug a PC into that cable..which means its normal ethernet.
More than likely that router will be more than fast enough. The main bottleneck now days is latency not bandwidth and you can't fix latency.