the price varies based on the part of the town you're looking to rent in. Usually, for 2-bedroom apartment with energies (waste, electricity, etc.) it will start at around 450 EUR (550 USD), but can go much higher if you choose a good location and bigger apartment. For 600 EUR you should be able to find a solid one, with luxurious ones in good location starting at 750 EUR a month.
Good parts of the town depend also on your work, etc. In general it is quite safe (as a rule of a thumb - historical center is the safest and most lucrative, the further you go, the less lucrative and cheaper it gets). The districts are:
1. Old Town (Staré Mesto) - pricey in general, but you can find some really nice apts for good price. Traffic is pretty ok although there's a lot of tourists in the historical center at times. You can also find everything within walking distance.
2. Ruzinov, Vrakuna, Podunajske Biskupice - first one is very nice, residential, with a lot of business centers in some locations. Traffic pretty ok, good connections with tram, most of necessities within walking distance as well. You can also find awesome stuff in Vrakuna and Podunajske Biskupice, but they're a bit further away and some parts can be shady, so you'd definitely have to ask some locals about the hood.
3. Nove Mesto, Raca, Vajnory - NM is pretty solid, in general a newer part of the town, but the traffic can get pretty bad in peak hours. Raca is quite beautiful in parts, but traffic is terrible in peak hours (although there are tram connections as well). Vajnory are further away from city center, so if you don't have a specific interest in that location, you should avoid it.
4. Karlova Ves, Dubravka, Lamac - KV is quite ok, but very crowded and traffic also tends to get pretty bad. Dubravka and Lamac are a bit further away from the center, but you can find good apts for very good price (tram works as well). (There are 3 other parts in this district - Devin, Devinska Nova Ves, Zahorska Bystrica. They're gorgeous, but far away from the center, with problematic traffic).
5. Petrzalka (Jarovce, Rusovce, Cunovo) - a project from the socialist era, Petrzalka is the biggest borough in Slovakia (some 100k people). Some people despise it, some love it. You can find awesome places there as well, but you'd have to see it. The only problem is the traffic, as to get to other parts of town you have to cross the Danube river using one of the 3 (4) bridges. 3 are for all types of traffic, 1 is for bikes, pedestrians, and trams only. If there's an accident, you're screwed. The three boroughs in parentheses are suburbs - again, very nice, but very far.
Some more info here as well: https://en.wikipedia.org/wiki/Boroughs_and_localities_of_Bratislava
There's a couple of real estate agencies that specialize on expats, but tend to have pricier apartments (e.g. http://www.expatexperts.sk/). Then there's general search engines (e.g. http://www.reality.sk/, https://reality.sme.sk/) and classifieds - think Craigslist - https://reality.bazos.sk/prenajmu/byt-2-1/. All are in ENG or you should be able to translate with Google in Chrome with acceptable results.
Couple of useful phrases just in case:
2-bedroom apartment - 2-izbový byt
Rent - prenájom
Apartment - byt
Price - cena
Country - štát
City - mesto
Floor area - podlahová plocha
Real-estate agency - realitná kancelária
Local classifieds - privátna (súkromná) inzercia
Good luck with your search!