SRM 4 consists of two different tracks: "Buried Underground", featuring the Orc Underground, extremist violence against its inhabitants, and lots of political maneuvering around it; and the "Artifact Rush", which is about artifacts, arcano-archeologists, and various events surrounding that.
For this question, I was mostly interested in the "Buried Underground" track, but I'll also briefly deal with the other half.
While timelines can of course be changed, SRM4 is originally intended to take place over the course of 2073 and 2074. Although the 12 missions in SRM4 makes it one of the biggest SRM seasons, that's still only one mission per two months, so some additional adventures (Sprawl Wilds and Elven Blood are very suitable) are a good idea to fill the gaps. Though you could also just ignore the timeline and condense it.
There are a couple of adventures outside SRM 4 that are related to the events in SRM 4:
- Splintered State is related; it features some of the same characters and themes, especially related to the political side of Buried Underground, also tying into the later SR5 metaplot about CFD.
- Copycat Killer is pretty much mandatory to run before you run SRM 04-05 On a Silver Platter. Takes place in January 2073, right at the start of SRM4.
- Carbon Copy, from Sprawl Wilds, which is basically a remake of Copycat Killer, changed to take place after instead of before SRM4, tying into the later CFD metaplot. (I think it would get too repetitive to run this if you've also ran Copycat Killer, and yet it refers back to Copycat Killer, which makes me not a fan of this.)
- Ashes, from Sprawl Wilds, featuring the first major attack on the Orc Underground, which several missions in SRM4 refer back to. Except the version in Sprawl Wilds has been changed to take place after the events in SRM4, which is a bit weird. The unaltered original is not available, but it's not hard to change it to make sense before SRM4; just remove the references to prop 23. According to most references, it takes place in the summer of 2072.
- Humanitarian Aid, from Sprawl Wilds. SRM 04-10 Romero & Juliette (from the Artifact Rush track) is basically a sequel to Humanitarian Aid. Romero & Juliette is also the most hated mission of SRM4, so it's fine to skip these. Or keep Humanitarian Aid but skip Romero & Juliette. But if you run it, run Humanitarian Aid first.
- The entire Artifact Rush track is kind of a follow-up to the 4 Dawn of the Artifacts adventures that take place about a year before the start of SRM4. I think part 2 takes place in the early months of 2072, though this is easy to move around.
- Artifacts Unbound also a follow up to Dawn of the Artifacts and ties more strongly to the Artifact Rush track. If you want, it culminates in a big event in July 2073.
- Dirty Tricks has more details about all the elections in 2074, including the referendum for prop 23 that's the focus and climax of the second half of Buried Underground. Storm Front also has some details including setting some actual dates to some of the adventures (which I'll include later in this answer).
- Free Taiwan, a free mission, uses many of the same recurring characters as the Artifact Rush track.
Good starting adventures for SRM4 are Ashes, Copycat Killer, and SRM 04-00 Back in Business. All three introduce the Orc Underground and various recurring characters. I think the chronological order: Ashes -> Copycat Killer -> Back in Business is best, but you could change this.
SRM4 should probably just be played in that order. You can easily drop all of the Artifact Rush missions if you want to focus on the Orc Underground, or vice versa. Back In Business introduces both tracks.
Copycat Killer should absolutely be played before On A Silver Platter. The latter makes little sense without it.
A character from SRM 04-00 Back in Business gets referred to in SRM 04-01 Hiding in the Dark and SRM 04-07 Burn or SRM 04-09 Assassin Nation, so it's best to keep that order.
The last couple of missions from Buried Underground, SRM 04-07 Burn, SRM 04-09 Assassin Nation, and SRM 04-11 Election Day, represent rising tensions and refer to each other a lot, so keep that order.
If you want to stick to "official" dates for the Buried Underground Missions, Storm Front puts some dates to this (though it's possible some other publication contradicts these dates):
- January 2073: Copycat Killer
- May 2073: Hiding in the Dark
- September 2073: Rally Cry
- January 2074: On A Silver Platter
- May 2074: Burn
- September 2074: Assassin Nation
- November 2074: Election Day
Of course you should feel free to ignore these dates and use what works best in your campaign. (In mine, I put On A SilverPlatter in July 2073 because I felt it was time to revisit the Orc Underground after a bunch of extended side adventures.) But you may have to adjust later references to these dates, because a lot of stuff refers to previous events.
I don't know the Artifact Rush track nearly as well, so I could be wrong, but my impression is that most of those missions stand more on their own than the Buried Underground ones, but there are still recurring characters.
If you want to focus on the Artifact Rush track but want to put it in a bigger context, you could check out Dawn of the Artifacts and Artifacts Unbound, but they're poor introductions; in terms of power level, they should be sequels; the runners meet a lot of powerful movers and shakers of the Shadowrun world, whereas the Missions are much more local, starting-level adventures. (I used part 2 and 3 of Dawn of the Artifacts and a mash-up of some Artifacts Unbounds adventures after the first couple of SRM4 missions to create some space to the later adventures; I'm not sure this was a good idea, because the scope and power level are very different.)