The Normal Procedure

You can buy Manchester United tickets on eticketing and the derby game here particularly.

You can buy Manchester City tickets direct from the Manchester city site, however MC have not yet started sale for next seasons matches (dates are yet to be confirmed).

Why that probably won't work

Note that the derby (MUFC vs MCFC) is a big deal in Manchester (source: I live there) and the derby match will be very popular (and only happens twice a year), although one could argue that Manchester United vs Liverpool is both local enough and has enough history to count as a third derby match. As such, tickets will be in very short supply. Note that while it is possible to buy tickets for other MUFC matches, it is only possible to apply for a chance to get United vs City tickets from Manchester United, and even then only if you join their supporters club (although you can do that online as a foreigner).

What else you can try

• Buy privately online: Sites like StubHub and GetMeIn provide an exchange service for people who hold tickets to sell them to those who want them, often for massively inflated prices. Note that while these services provide some protection, fake tickets are also sold here. Ebay is another option, although there watch out carefully for previous year or fake tickets.
• Stump up the cash for a hospitality package. Manchester United will sell you a "box" (a private space with "table service" attendance etc) for the United vs City match for £5000 for 5 people at the moment.
• Book a travel agent package. ~~See the Manchester City options with Thomas Cook here and the Manchester United/Thomas Cook here~~. Note that the derby game again has special restrictions applied to it. Also, you'll have to buy more than just the ticket with the travel agent.Edit Autumn 2019: With the collapse of Thomas Cook, there does not currently appear to be a travel agent offering "official" packages, although MCFC do provide this page
• Go to a non-derby game. Tickets are still in demand, but they are available, and tourists to buy them.
• Try and get a ticket as a visiting fan. This won't help you for derby matches, but it may work for some others
• Hope that United and City meet in one of the knock-out tournaments (unlikely, and even more unlikely that it will happen to coincide with your trip, but it may occur). The two that they are both in are the Capital One (formerly the League) Cup and the FA Cup.
• Trying on the day. Near the grounds (although out of sight of security) you may well find "Touts", people who buy and sell tickets for profit. It's also sometimes possible to find in local pubs frequented by fans, people who bought tickets for friends who failed to show up. You are likely to pay inflated prices this way, with no guarantee of getting a ticket at all - but it is an option.
• Seeking out hospitality options. Lots of companies hold season tickets to use as part of marketing to clients. You may have connections that would allow you to make use of these (especially as some are quite undersubscribed. My partners former work use to hold 2 seats that were frequently only given/sold in the days immediately before a match).

If it doesn't have to be Manchester United or Manchester City

Manchester is fairly central to one of the most populated areas of the UK (the band roughly from Liverpool to Hull) and has good (if busy) transport links. There are lots of other places you could go to watch football:

Premier League

• Everton or Liverpool. Both based in Liverpool, about 1 hour train journey, plus travel to the ground.
• Burnley. A Lancashire team, not far from Manchester. About 50 minute train or 45 minute taxi

Lower Leagues

You can find teams from these anywhere, so I'll limit myself to teams in Greater Manchester that may be of interest:

• Stoke City. Tier of play: 2 About 35 minutes train journey to the city of Stoke on Trent + transfer to the stadium. Were last in the Premier League 2017-2018
• Bolton Wanderers. Tier of play: 3. Bolton is a town to the north of Manchester proper. The Wanderers were in the Premier League as recently as 2011-2012
• Bury. Tier of play: 3. Another town to the north of Manchester, this one served by Metrolink (the tram) service. Not played inTier one since 1928
• Oldham Athletic Tier of play: 4. This town is to the east of Manchester, again served by Metrolink. Were last in the Premier League 1994-95.
• Rochdale. Tier of Play: 3. Town to the North East of Manchester, again served by Metrolink (although it's a long journey by tram)
• Wigan Athletic. Tier of play: 2. Town to the North-West of Manchester, played in the Premier League as recently as 2013
• Salford City. Tier of play: 3. The "other city" in Manchester. Owned by former MUFC players.

There are many more to be found in the Wikipedia article. Salford City and F.C. United of Manchester both have links to Manchester United. (Salford part-owned by former players, FCUM owned by former fans)/