Good advise here thus far, if you are unclear what your rigs configuration is, look in the glove box if still legible, there should be a sticker with the RPO codes that will tell you what options your Burb came with, multiple RPO code converters on the web to put cod to a description what it means.
I have a 98 C1500 with the 6.5 Diesel some of the same consideration will apply to yours even though yours is a gasser.
@ 30 years old everything is suspect age aside you don't know what a previous owner(s) has/have done in abuse, upgrades, or maintenance so front to back check it all out.
Change all fluids/filters you didn't indicate C or K 1500 if K often overlooked is fluid change in the Xfer case, a deep trans pan with extra capacity is good addition, I run the PML pan in mine aluminum aids in heat xfer and add 3.75 qts..
Does your 5.7 have the rigid aluminum push clip oil cooler lines into a oil in grille cooler (usually part of the "towing package") if yes on the 6.5 hat is a bad fail point if they fail you lose oil pressure & engine quickly if at highway speeds, maybe consider something aftermarket there if your is the same.
For braking I went with SSBC twin piston calipers & slotted drilled rotors up front, OK upgrade there, replaced my vintage rubber brake lines with braided stainless/teflon inner to preclude collapsed inner hose liners, I also did a rear drum to disc conversion, I probably would not do that again I think the drums stop as well as the discs, I might do a complete rear axle swap that is already configured from factory as disc, but I'm good for now.
Check out Banks rear diff cover, I'm running PML there as well, Banks current rear diff cover wasn't avail back in early 2000s, would be a good opportunity to inspect diff gears and refresh with new diff oil.
I also run a Derale electric fan, Temp switch option controlled aux trans cooler https://derale.com under the truck, which keeps things cool especially in Summer stop/go traffic and you are not getting much cross flow through the radiator/AC condenser, also a little more system fluid capacity for tans in the cooler & plumbing to/from the cooler. Side benefit the engine runs a littler cooler as well, since the trans cooler section of your radiator is not competing/inducing heat with/to coolant in the rad as it tries to keep engine temps in check.
On topic of radiator, maybe a fresh T Stat and coolant change, remove radiator and deep clean the fins, between AC cond. & Rad. is a negative pressure area you will be amazed what gets sucked in there, bugs-dirt, I had a little bird carcass in mine. By deep clean I mean get some foamy AC coil cleaning solution spray the fins, let soak, 30 min or so, rinse with garden hose & nozzle set on fan or rain patterns, don't ue solid full force stream or you will bend over the fins. I do mine by connecting hose to my water heater drain, let it flow no nozzle too get crud out the water heater, then put nozzle on it and use clean warm/hot water to rinse foam/dirt out of the radiator fins. Foam rinse repeat until no more crud coming out, do the same for AC condenser, you should be able to leave it in the truck so you don't have to let charge out, word of caution the foam is a chemical so may remove old paint, so be careful when applying and rinsing.
How old is water pump, fan clutch, and hoses know/replace BEFORE you go hooking up to a trailer/load, how about your fuel tank is it clean in the bottom, maybe a new fuel pump in tank would be in order, is the filtration "sock on the pickup tube crudded up?
For springs in rear your might want a new set especially if past life, the Burb did a lot of towing, on my 98 K1500 truck also with the 6.5 Diesel, (it can/has tow(ed) VERY big many mods also) since 2002 it has had the Road master XXF kit on it https://activesuspension.com/pages/ras I prefer it over bags, as it is passive, you run on standard suspension ride unloaded, springs "engage" as load is added, it helps with trailer sway, and anti-torque "axle wrap".
If you have a good trans shop you trust, have them take your Burb on a "check ride" with scan tool hooked up so they can watch pressures/shifting, maybe a towing shift kit/valve mod would be worthwhile.