How do Y.P and welcome.
Now, with EVERY respect to Geko, my personal opinion is that you should jump straight into computer based recording using an Audio Interface and Digital Audio Workstation, the latter is the software used for the recording/play process. Now, I am going to assume you have a computer of some sort? It really does not matter what, anything made in the last 5 years or so that can run Windows 7 or later will be fine for what is a pretty low resource hit. Next you will need..
1) Audio Interface. I shall suggest ONE, upfront to save you searching! Steinberg UR22.
2) A microphone. WHOAH! Lot of opinions here but, 'trumpet' you say? I guess a capacitor(aka 'condenser) microphone is used but for now I would stick to a dynamic and unless someone here knows better, the Shure SM57. A stand of course and an 'XLR to XLR' microphone cable. The mic could also be used on the Fender amp* but bass is often better recorded by 'Direct Injection' The Steinberg can do both at the same time!
3) DAW. The UR22 comes with a copy of Cubase, a first class piece of software but many find it daunting to learn (I have it, never did grasp it well!) There is a DAW called Reaper. This is free to try until you feel you orta pay but it is only $60.00 Less steep a learning curve than Cubase but NO DAW is easy.
4) Monitoring. Since you will be 'playing along' you will need headphones. Again a VAST choice but two fundermantals. For you immediate purpose they must be of the 'closed back' type to prevent their sound 'spilling' into the mic and secondly, almost regardless of the sound quality they must be comfortable for extended periods of wear. As it happens I have just bought some AKG K92s and I love 'em but, as I say, lots out there.
Really accurate monitoring for mix down purposes needs monitor speakers (tho' a decent hi fi rig will serve at first) but anything decent that can handle YOUR choice of instruments is going to be bloody expensive! I mean very little change out of $2000 per pair.
If that list has frightened you off? Good. Not being mean but what you want to do will need the outlay of quite a bit of cash, around $4-500 and a significant amount of time and effort.
*I will hunt up the manual for that amp and see if it has any useful facilities.
Dave.