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My understanding is that when my computer wants to resolve a host name, it will ask the DHCP-assigned DNS servers first. If those don’t know, it goes and asks the ‘root servers’ who then defer to another server, etc.

Using dig or a similar tool, is it possible to see the string of servers that get asked, in a similar way as traceroute shows how packets travel to a certain destination!

As others have already answered, most usually a host sends a recursive query to a nominated resolver, often a local server or router, often belonging to an ISP or Google's well-known (distributed) 8.8.8.8 and 8.8.4.4. Typically it will have a couple defined, and if the first doesn't respond, it will move to the next after a timeout.

Name resolvers are free to use whatever algorithm they like, including just ask for NS records (for com., stackexchange.com. right from the top) then an A record. And also caching, within responded time-to-live parameters.

To find out what a particular name resolver does you'll need to monitor packets or have access to the server.

But you can certainly do it manually if you're trying to follow a chain, such as for debugging broken delegations, missing glue records and so on.

To resolve the A record of www.stackexchange.com:

Find a root zone server from the list, which every resolver will have installed:
http://www.internic.net/domain/named.root

We pick the first one, 198.4.10.4

Ask who to ask:

nslookup -norecurse -query=ns www.stackexchange.com 198.41.0.4

Root server says "Don't ask me, ask com questions ate.gtld-servers.net", and helpfully gives the address:

Authoritative answers can be found from:
com nameserver = e.gtld-servers.net.
e.gtld-servers.net internet address = 192.12.94.30
(other answers trimmed)

Ask e.gtld-servers.net who to ask:

nslookup -norecurse -query=ns www.stackexchange.com 192.12.94.30

He says "don't ask me, ask stackexchange.com questions at ns-925", and helpfully gives the address.

Non-authoritative answer:
*** Can't find www.stackexchange.com: No answer
Authoritative answers can be found from:
stackexchange.com nameserver = ns-925.awsdns-51.net.
ns-925.awsdns-51.net internet address = 205.251.195.157
...

We see that ns-925 is nameserver for the domain we're after (stackexchange.com) so we ask for the A record:

nslookup -norecurse -query=a www.stackexchange.com 205.251.195.157

He says gives a CNAME response:

www.stackexchange.com canonical name = stackexchange.com.

But as it's a nameserver for that as well, it (helpfully) gave us the the A record of the CNAME record:

Name: stackexchange.com
Address: 151.101.129.69

A normal DNS client just queries the (DHCP or statically) assigned DNS servers. Only DNS servers usually query root servers, doing a recursive query.

You can use a packet capture of your choice and filter for UDP (TCP) source or destination port 53 to see the communication with any DNS server.

Which servers and exact method a DNS client uses is up to the OS or the application and off-topic here.

I think you are looking for dig +trace. The following command

dig +trace networkengineering.stackexchange.com 

will query the root name servers (NS) for the NS of .com, then the .com NS for the NS of stackexchange and finally will get an answer for networkengineering.