The easiest rule of thumb is to follow the circular avenues you can clearly spot on maps. There are three main circles, if you take the red metro line (M1) as a reference, the metro crosses the first circle between San Babila and Palestro station, the second circle at Porta Venezia station and the outer circle at Loreto station.
Someone drew a funny map a few years ago where the first two circles were labelled as "Serie A" and "Serie B": think of Westminster/Chelsea for the first circle and London zone 1 for the second circle. Between the second circle and the outer circle is where most of the middle class (or what is left of it) live and ideally where you want to find a place: 99% of this area is safe, has good transport connections and a variety of shops and venues. I'd say think London zone 2. My personal favorites would be Porta Romana, Isola or Porta Genova but really you can't go wrong if you stay inside the outer circle, also called the "90/91 circle" from the number of the bus lines that follow exactly that route, line 90 going clockwise, line 91 counter-clockwise. They're also the only two transport lines running 24 hours 7 days a week, usually overcrowded but very convenient to come back home late at night without paying a taxi.
If you want to venture a little bit further (and you could be forced to, given the crazy prices of Milan hotels), the areas outside the outer circle are a bit like London zone 3/4: some terrible districts and some very nice ones. Some areas I would consider are Città Studi (metro station: Piola), and Lambrate in the North-east, also Gorla and Istria in the North (both with convenient metro stations), Viale Corsica in the East (S-line station: Porta Vittoria), Bande Nere in the West.
I don’t have any first-hand experience with specific hotels, but I know there are two Ibis with (usually) reasonable prices: one right in the city centre and a cheaper one in the North, near the newly opened “Ca Granda” metro station. This used to be a bit remote but now with the new M5 metro line it’s roughly 30-40 mins from the centre and the neighbourhood is a non-trendy, quiet residential area.